What is the basis of the Spanish claim to Gibraltar?

Different interpretations of the Treaty of Utrecht are one point on both sides, but they aren't the main point for any of them.

For Spain, Gibraltar is a colony. Colonies are the dominion of a country over another one or a part of it, and even if it was acquired under a treaty, colonization is no longer a right in the XXI century and the colonized country has the right of get back its independence and its territory. Even if the transfer of Gibraltar under the Treaty of Utrecht were valid, the situation there is not seen different than any other British colony (like Malta, Kenya, Zimbabwe or India, for example) that in spite of having been (more or less) legally acquired, had right to independence under the people that originally inhabited it. Furthermore, the original inhabitants of Gibraltar and its institutions left in 1706 and settled in nearly San Roque, which is still the "Very Noble and Very Loyal city of San Roque, where Gibraltar lives on" (Spanish: Muy Noble y Muy Leal ciudad de San Roque, donde reside la de Gibraltar).

On the British side, their stated basis to their claim on Gibraltar is the will of its inhabitants, which nearly unanimously want to keep being a British colony and reject joining Spain. In fact, the United Kingdom seem to have embraced self-determination as the very reason to keep its remaining colonies, as it could be seen in a 1982 related case: “The Falkland Islands are once more under the government desired by their inhabitants. God save the Queen.” Interestingly, the Spanish government has always rejected right of self-determination to any part of Spain and under that perspective Gibraltar is not different from Catalonia or the Basque Country.

Furthermore, there are differences of interpretation about what was ceded on the Treaty of Utrecht. For the British, Spain ceded the sovereignty of the city of Gibraltar, including the whole peninsula, territorial waters and so on. From the Spanish perspective, it was just about the property of the city and the military use of the port - that is, Gibraltar is legally just like a British military base in Spanish soil. However, claims grounded on the Treaty of Utrecht are belittled by claims about self-determination and decolonization.

Interestingly, a good deal of Spanish and British basis to claim Gibraltar could be reversed in Olivença, Ceuta and Melilla, just seen by Spain as Spanish cities.

Full text of the Article X of the Treaty:

The Catholic King does hereby, for himself, his heirs and successors, yield to the Crown of Great Britain the full and entire propriety of the town and castle of Gibraltar, together with the port, fortifications, and forts thereunto belonging; and he gives up the said propriety to be held and enjoyed absolutely with all manner of right for ever, without any exception or impediment whatsoever.
But that abuses and frauds may be avoided by importing any kind of goods, the Catholic King wills, and takes it to be understood, that the above-named propriety be yielded to Great Britain without any territorial jurisdiction and without any open communication by land with the country round about.
Yet whereas the communication by sea with the coast of Spain may not at all times be safe or open, and thereby it may happen that the garrison and other inhabitants of Gibraltar may be brought to great straits; and as it is the intention of the Catholic King, only that fraudulent importations of goods should, as is above said, be hindered by an inland communications. it is therefore provided that in such cases it may be lawful to purchase, for ready money, in the neighbouring territories of Spain, provisions and other things necessary for the use of the garrison, the inhabitants, and the ships which lie in the harbour.
But if any goods be found imported by Gibraltar, either by way of barter for purchasing provisions, or under any other pretence, the same shall be confiscated, and complaint being made thereof, those persons who have acted contrary to the faith of this treaty, shall be severely punished. And Her Britannic Majesty, at the request of the Catholic King, does consent and agree, that no leave shall be given under any pretence whatsoever, either to Jews or Moors, to reside or have their dwellings in the said town of Gibraltar; and that no refuge or shelter shall be allowed to any Moorish ships of war in the harbour of the said town, whereby the communication between Spain and Ceuta may be obstructed, or the coasts of Spain be infested by the excursions of the Moors. But whereas treaties of friendship and a liberty and intercourse of commerce are between the British and certain territories situated on the coast of Africa, it is always to be understood, that the British subjects cannot refuse the Moors and their ships entry into the port of Gibraltar purely upon the account of merchandising.
Her Majesty the Queen of Great Britain does further promise, that the free exercise of their religion shall be indulged to the Roman Catholic inhabitants of the aforesaid town.
And in case it shall hereafter seem meet to the Crown of Great Britain to grant , sell or by any means to alienate therefrom the propriety of the said town of Gibraltar, it is hereby agreed and concluded that the preference of having the sale shall always be given to the Crown of Spain before any others.

Caveats:

1. I believe this question fits more politics.SE than history, since it asks for the current claims. Nonetheless, since nobody else has raised this point and it already has an answer, I will not try to migrate it.

2. Without disagreeing with the meaning of Samuel Russell's answer, territorial claims are accompanied by some reasoning, as they are needed to convince public opinion (both internal and external) to support that claim (if you did not need public opinion to take control you would not make a claim, you would just take control of the disputed land). Of course, that someone makes a claim does not mean that you have to agree with it or even that it makes sense.

3. I am not expert in 18th century laws. I am giving some explanations (and criticism) for the points as they have been explained to me, please do not take it as endors*m*nt of any position as I do not want to engage in a debate.

Main points:

1. The treaty of Utretch states that that the above-named propriety be yielded to Great Britain without any territorial jurisdiction and without any open communication by land with the country round about.¹ is interpreted by some people to mean that what the UK got was in essence some Spanish real state; a piece a land that was part of Spain but owned by the UK.

   Now, we are dealing with a treaty from a time where countries were more or less the King's land to give and take away; the argument would be rather convincing if it were in a modern treaty (where ownership by a state is very different from sovereignty), but that could be not easy to apply to a 18th century document. Also, it contradicts the previous and he gives up the said propriety to be held and enjoyed absolutely with all manner of right for ever, without any exception or impediment whatsoever.

2. Similarly, Spain believes Gibraltar was taken in the context of a Spanish dynastic dispute; the point here is that the Succession War was about who would be King of Spain and not about some UK right on Gibraltar; this argument would interpret the Treaty of Utretch as kind of a "private deal" between the Kings involved that harmed Spanish sovereignty.

   Again, to that you could counter that maybe that argument would be anachronistic when dealing with a 18th century treaty. YMMV.

3. Gibraltar is a colony and the UN Charter condemns colonialism and calls for its end². Well, Gibraltar is not governed by the same UK law, has no vote in internal UK politics, is far away from the metropoli, so it is not much of a stretch to consider it a colony.

   Spanish position is that, Gibraltar not being a country, then history, geography and even the wording of the treaty (the "without any jurisdiction" clause, and the fact that Spain gets the first dibs to repurchase Gibraltar) means that the treaty recognizes Gibraltar as part of Spain for the decolonization process.

   There you could trace a paralel with the Portuguese colony of Goa in India, for example³. At the time of decolonization, Goa had been a Portuguese colony for many centuries, and had never been a party of "India" (which did not exist as a country before the Portuguese took Goa).

4. A minor point is that the current territory of Gibraltar exceeds that agreed upont the Treaty, with Gibraltar having expanded upon a strip of land that the Spanish left unoccupied in order to have a "neutral zone".

² I do not know the legal force of that declaration; if it is binding of it is a declaration of intentions further developed in other instruments.

³ While, of course, noting that India did not get Goa back through a recognition of their claims.

1) what is the basis for the Spanish state’s actions? Any damn basis they please. They’re a Westphalian state. Being a sovereign entity involves never having to say you’re sorry unless another entity has a greater force of arms or you’ve voluntarily bound yourself. And making ridiculous unenforced claims tends to be a capacity that other entities don’t take armed action over.

2) regarding the basis for the claim in particular, ethnic-lingual nationalism is a common basis for claims of this sort

3) The British state currently possesses a capacity to close a major shipping route easily. Spain would benefit from removing this, thus enhancing its current capacity to do likewise

4) Utrecht gave Spain first purchase option on disposal. This does indicate that the states observing that treaty believe Gibraltar to have a connection to the inheritor Spanish state party to that treaty

So, please, what is the basis for Spain's territorial claim?

You seem to be under the impression that states need to act as rational agents with justified beliefs. This is demonstrably not the case. As you observe irrational claims are unlikely to be persuasive with international bodies. That Gibraltar is still a British overseas territory indicates that such claims have not yet been persuasive.