How were 19th century American diplomats paid?

They were paid a regular salary and given an "expense account" of sorts. At least, the higher ranking representatives of the United States were. While this was probably not a very adequate amount, American ministers were definitely not expected to pay for everything out of their own pockets.

Early United States ambassadors were paid around $2,500, while consuls were unpaid appointments who receive remunerated from fees. That was of course completely inadequate, and the latter eventually resulted in abuses all around. As the federal government became better organised by the early 19th century, diplomatic salaries had been set to:

• Ministers Plenipotentiary: $9,000
• Ministers Resident: $6,000
• Chargés d'affaires: $4,500
• Secretaries of Legation: $2,000

_{(Source: Sparks, Jared, Francis Bowen, and George Partridge Sanger. The American Almanac and Repository of Useful Knowledge for the Year 1843. Vol. 31. Boston, David H. Williams, 1860)}

Additionally, the ministers and chargés may receive an additional expense allowance ("outfit") of up to (at Presidential determination) one year's salary. It seems that a practice developed where half of their annual salary was received for outfit when a diplomat is first posted, and a quarter when they returned. This would presumably go towards the upfront expenses of moving, accommodations, etc.

These numbers did not change much for many subsequent decades. The same figures appeared in references up till the 1840s and 50s. At some point however they were raised such that, by 1872, the pay for top American diplomats were about:

• Ministers Plenipotentiary: $17,500¹ or $12,000²
• Ministers Resident: $7,500³

Notes:

1. To the United Kingdom, the German Empire, the French Republic
2. To Austria, Mexico, Russia, Spain, Italy, China, Brazil
3. With the exception of Liberia, $4,000

_{(Source: Turner, A.J., Legislative Manual of the State of Wisconsin, Madison, Wis.: Atwood & Culver, 1872)}

While diplomats represented government interests, consuls were also appointed to countries to assist Americans locally. They remained unpaid by salary until the 1850s, when rules were introduced to curb the excess. But by 1886:

A very careful investigation on [cost of living] was made by the British consuls, by order of their government, in 1873. Notwithstanding this advance of prices, the scale of salaries of American consuls has scarcely been changed in these thirty years, when men are still sent to Florence and Naples, expected to be competent to perform all the duties of the office, hold a respectable position in society, and pay proper attention to the numerous Americans visiting those places, on the sum of $1,500 a year, scarcely more than is paid to the subordinate employees of Congress.

(...)

[I]t may be presumed that $2,500 is considered by Congress high salary. The experience of every one who has ever been in the consular service, or who has lived abroad in a private capacity, shows that in most cases this is utterly insufficient.

_{- Schuyler, Eugene. American Diplomacy and the Furtherance of Commerce. New York, 1886.}

During this period, the pay were about:

• Consuls-General: $2500 - $6000
• Salaried Consuls: $1,500 - $4,000^☨, and cannot engage in business
• Consuls: $1,000, but may engage in business
• Feed Consuls: paid by consulate fees up to $2,500, and may engage in business

_{☨: Except the Consuls to Liverpool and Hong Kong, who were paid $6,000 and $5,000, respectively.}

That salaries were inadequate and fixed for long periods despite rising costs were a continuous issue. For example, in 1816 the American Ambassador to the Court of St James's, John Quincy Adams, wrote to then Secretary of State James Monroe that:

During the five years and a half of my establishment at St. Petersburg, my expenses fell little short of but did not exceed the salary and outfits allowed me. But I certainly could not disguise to myself ... that it was impossible to proportion my establishment to that standard without a great sacrifice of that consideration which attends the character of a foreign minister.

I may state with perfect confidence that no minister of the United States at this court has ever found it profitable to limit his expenses within the public allowance of salary and outfit. And while it is notorious that a salary fixed twenty-five years ago was then inadequate to the necessities of the station, it is equally notorious that every expense fo a domestic establishment in this country has doubled in that interval.

_{- Adams, John Q. "To the Secretary of State." Letter to James Monroe. 12 July 1816. London.}

Europe

Not directly related to the question, but European states did not rely on the personal wealth of diplomats, either. Or at least, not exclusively. In the same letter to Monroe, Adams explained that in addition to a base salary:

[T]he allowances to them for contingent expenses are usually an additional expense equal in amount to the salary. They are also entitled after a few years of service to pensions for life, proportioned to the length of time they have been in service, and equal upon the average from one-third to one-half of the annual salary, and they are permitted to receive presents from the governments to which they are accredited, which in these treaty making times form no inconsiderable part of their emoluments ... The Russian and Austrian governments pay their ministers abroad much upon the same scale as France and England. The Russian ambassador at this court has a salary of sixty thousand dollars and a house to live in rent free.

_{- Adams, John Q. "To the Secretary of State." Letter to James Monroe. 12 July 1816. London.}

Compared to the Russian ambassador to Britain, the President of the United States at the time received a paltry $25,000. While Adams was advocating raises for diplomats like himself, there's little doubt that European governments supported their representatives well. For example, Britain's ambassadors enjoyed expense accounts to the sizes of:

English ministers also have an outfit in order to enable them to install themselves properly at their posts. This is calculated on a liberal scale, being, for example, $20,000 for Paris; $12,500 for Vienna, Berlin, and St. Petersburg; $10,000, for China, Japan, and Persia, Madrid and Washington, and never being less than $2,000.

_{- Schuyler, Eugene. American Diplomacy and the Furtherance of Commerce. New York, 1886.}