The land borders are currently restricted for entry from Mexico into the US until 8 November, 2021.

(travel INTO Mexico continues unrestricted as normal)

From 8 November, 2021, fully vaccinated travelers may enter the US via the land borders for non-essential travel.

From January 2022, both essential and non-essential travelers must be fully vaccinated to enter the US via the land borders.

Source – there were a series of Tweets by the Department of Homeland Security on 20 October, 2021:

Tweet 1:

As previously announced, starting Nov 8, travelers from Mexico and
Canada who are fully vaccinated for #COVID19 may enter the U.S. for
non-essential travel via land and ferry ports of entry.

Tweet 2:

The timing of this change is aligned with the new international air
travel system that will also be implemented in November.

Tweet 3:

Restrictions continue to apply to non-essential travelers who are not
fully vaccinated.

Tweet 4:

Beginning in January 2022, both essential and non-essential travelers
from Canada and Mexico must be fully vaccinated for #COVID19 and
provide related proof of vaccination to enter the U.S. via land or
ferry ports of entry.

Shortly after asking this question I found the answer in a tweet by Chad Wolf, the acting secretary of Homeland Security:

We continue to work with our Canadian and Mexican partners to slow the
spread of #COVID19. Accordingly, we have agreed to extend the
limitation of non-essential travel at our shared land ports of entry
through September 21.

I also found this Fact Sheet on the DHS website that also links to the same tweet, so it looks like this tweet is the official announcement.

So for now the borders are still restricted until at least Sept 21. The list of exceptions for the border closure is as follows:

• U.S. citizens and lawful permanent residents returning to the United States;
• Individuals traveling for medical purposes (e.g., to receive medical treatment in the United States);
• Individuals traveling to attend educational institutions;
• Individuals traveling to work in the United States (e.g., individuals working in the farming or agriculture industry who must travel between the United States and Mexico in furtherance of such work);
• Individuals traveling for emergency response and public health purposes (e.g., government officials or emergency responders entering the United States to support federal, state, local, tribal, or territorial government efforts to respond to COVID-19 or other emergencies);
  Individuals engaged in lawful cross-border trade (e.g., truck drivers supporting the movement of cargo between the United States and Mexico);
• Individuals engaged in official government travel or diplomatic travel;
  Members of the U.S. Armed Forces, and the spouses and children of members of the U.S. Armed Forces, returning to the United States; and
  Individuals engaged in military-related travel or operations.

So effectively speaking US citizens and permanent residents are not affected by these restrictions in any way. They may continue traveling freely in both directions unhindered.

Update Sept 18 – According to Facebook post by the Mexico Consulate in San Diego, the border restrictions will be extended for another month until at least Oct 21.

Mexico has proposed to the US the extension for an additional
month of the restrictions on non-essential land travel on our common
border, until Oct. 21, to avoid the spread of #COVID19. Restrictions
applied since March will continue in operation & will not interrupt
binational trade.

The DHS Fact Sheet has also been updated to include this extension.

Update Oct 19 – The border restrictions have been extended until at least November 21. Statement Tweet by Chad Wolf:

To continue to limit the spread of COVID, the US, Mexico, & Canada
will extend the restrictions on non-essential travel through Nov 21.
We are working closely with Mexico & Canada to identify safe criteria
to ease the restrictions in the future & support our border
communities.

The DHS Fact Sheet has again been updated.

Update Nov 20 – The border restrictions have been extended until at least December 21. Statement Tweet by Chad Wolf:

In order to continue to prevent the spread of COVID, the US, Mexico, &
Canada will extend the restrictions on non-essential travel through
Dec 21. We are working closely with Mexico & Canada to keep essential
trade & travel open while also protecting our citizens from the virus.

Update Dec 11 – The border restrictions have been extended until at least January 21. Statement Tweets by Chad Wolf and by Secretaría de Relaciones Exteriores de México:

In order to continue to prevent the spread of COVID, the US, Mexico, &
Canada will extend the restrictions on non-essential travel through
Jan. 21. We are working closely with Mexico & Canada to keep essential
trade & travel open while also protecting our citizens from the virus.

Las restricciones se mantendrán en los mismos términos en que se han
desarrollado desde su implementación el 21 de marzo.
Ambos países coordinarán las medidas sanitarias en la región
fronteriza que estarán vigentes hasta las 23:59 hrs. del 21 de enero
de 2021.

Translation:

Restrictions will remain in the same terms as their implementation was
developed on March 21.
Both countries will coordinate health measures in the border region
which will be in force until 23:59 hrs on 21 January 2021.

Update January 13 – The border restrictions have been extended until at least February 21. Statement Tweet by Homeland Security:

In order to continue to prevent the spread of #COVID, the US, Mexico,
Canada will extend the restrictions on non-essential travel through
Feb. 21. We are working closely with Mexico & Canada to keep essential
trade & travel open while also protecting our citizens from the virus.

Update March – The border restrictions have been extended until at least March 21. Facebook post by Mexican government:

Tras revisar el desarrollo de la propagación de #COVID19 y debido a
que diversas entidades federativas se encuentran en 🟠 del Semáforo
Epidemiológico, 🇲🇽 planteó a 🇺🇸 la extensión, por un mes más, de
las restricciones al tránsito terrestre no esencial en su frontera
común. Las restricciones se mantendrán en los mismos términos en que
se han desarrollado desde su implementación el 21 de marzo de 2020.
Ambos países coordinarán las medidas sanitarias en la región
fronteriza que estarán vigentes hasta las 23:59 hrs. del 21 de marzo
de 2021.

Translation by Facebook:

After reviewing the development of the spread of #COVID19 and due to
various federative entities in 🟠 of the Epidemiological Light, 🇲🇽
raised 🇺🇸 the extension, for one more month, of non-essential land
transit restrictions in their common border. Restrictions will be
maintained on the same terms as they have developed since their
implementation on March 21, 2020. Both countries will coordinate
sanitary measures in the border region that will be in effect until
23:59 hrs. of March 21, 2021.

Update 18th March – The border restrictions have been extended until at least 21st April. Statement Tweet by Homeland Security:

To prevent the further spread of COVID-19, and in coordination with
our partners in Canada and Mexico, the United States is extending the
restrictions on non-essential travel at our land borders through April
21, while ensuring continued flows of essential trade and travel.

And similar Tweets by Mexico – Tweet 1, Tweet 2

Para prevenir la propagación de #COVID19, Mexico impondrá a
partir del 19 de marzo de 2021, restricciones al tránsito terrestre
para actividades no esenciales en su frontera norte y sur.
To prevent the spread of #COVID19, Mexico will
impose as of March 19, 2021, restrictions on land traffic for
non-essential activities on its northern and southern borders.

Adicionalmente, el @GobiernoMX desplegará medidas de control
sanitario en el norte y sur del país.

Las restricciones al tránsito no esencial y las medidas sanitarias se
mantendrán vigentes hasta las 23:59 hrs. del 21 de abril de 2021.
Additionally, the @GobiernoMX It will
deploy sanitary control measures in the north and south of the
country.

Restrictions on non-essential traffic and sanitary measures will
remain in effect until 11:59 p.m. April 21, 2021.

Update April – The border restrictions have been extended until at least 21st May. See Tweet by US DHS:

To deter the spread of #COVID19 and protect our citizens, the United
States is continuing restrictions on non-essential travel at our land
borders through May 21, while maintaining the flow of essential trade
and travel as we have for over a year.

and a follow up tweet

We are guided by science and public health data and engaged in
discussions with Canada and Mexico about easing restrictions as health
conditions improve.

Update 20 May – The border restrictions have been extended until at least 21st June. See Tweet by US DHS:

To fight #COVID19 spread and protect our citizens, the U.S. is
continuing restrictions on non-essential travel at land borders
through June 21, while allowing essential trade & travel. We’re
working closely with Canada & Mexico to safely ease restrictions as
conditions improve.

Update 20 June – The border restrictions have been extended until at least 21st July. See Tweet by US DHS:

To reduce the spread of #COVID19, the United States is extending
restrictions on non-essential travel at our land and ferry crossings
with Canada and Mexico through July 21, while ensuring access for
essential trade & travel.

which was followed up by this tweet:

DHS also notes positive developments in recent weeks and is
participating with other U.S. agencies in the White House’s expert
working groups with Canada and Mexico to identify the conditions under
which restrictions may be eased safely and sustainably.

Update 21 July – The border restrictions have been extended until at least 21st August. See Tweet by US DHS:

To decrease the spread of COVID-19, including the Delta variant, the
United States is extending restrictions on non-essential travel at our
land and ferry crossings with Canada and Mexico through August 21,
while ensuring the continued flow of essential trade and travel.

and follow-up tweet:

DHS is in constant contact with Canadian and Mexican counterparts to
identify the conditions under which restrictions may be eased safely
and sustainably.

Update 20th August – Border restrictions for non-essential travel have been extended until at least 21st September. Tweet by US DHS:

To minimize the spread of #COVID19, including the Delta variant, the
United States is extending restrictions on non-essential travel at our
land and ferry crossings with Canada and Mexico through September 21,
while continuing to ensure the flow of essential trade and travel.

Update 22nd September – The border restrictions for non-essential travel have been extended until 21st October. Tweet by US DHS:

To minimize the spread of #COVID19, including the Delta variant, the
United States is extending restrictions on non-essential travel at our
land and ferry crossings with Canada and Mexico through October 21,
while continuing to ensure the flow of essential trade and travel.

which was followed by this tweet:

This Notification may be amended or rescinded prior to that time,
based on public health considerations.

Update October 2021:

From 8 November, 2021, fully vaccinated travelers may enter the US via the land borders for non-essential travel.

From January 2022, both essential and non-essential travelers must be fully vaccinated to enter the US via the land borders.

Source – there were a series of Tweets by the Department of Homeland Security on 20 October, 2021:

Tweet 1:

As previously announced, starting Nov 8, travelers from Mexico and
Canada who are fully vaccinated for #COVID19 may enter the U.S. for
non-essential travel via land and ferry ports of entry.

Tweet 2:

The timing of this change is aligned with the new international air
travel system that will also be implemented in November.

Tweet 3:

Restrictions continue to apply to non-essential travelers who are not
fully vaccinated.

Tweet 4:

Beginning in January 2022, both essential and non-essential travelers
from Canada and Mexico must be fully vaccinated for #COVID19 and
provide related proof of vaccination to enter the U.S. via land or
ferry ports of entry.