I'll defer to Andy Hall on this one-- he definitely has more of the background and the references in the Gulf area, especially Texas, than I do. But a few comments on the overall situation (some of which have already been mentioned, but I'll repeat for the sake of completeness):
The mouth of the Rio Grande presented a very thorny puzzle for the would-be blockader. Due to its being an international border, it could not be effectively (and legally) blockaded, because the U.S. could not blockade the port of a country with which it was not at war (we will leave all the nitpicking about whether the U.S. was at war with another country out of this for the time being-- from the international point of view, that was the de facto situation, if not de jure). Local conditions also made it really difficult-- the mouth of the Rio Grande is shallow, with lots of intervening shoals and little islands, some merely temporary, so a close blockade was really out of the question anyway. The only real solution for the North was to take and hold Brownsville and some of the other principal points, but supplying such a a far-flung garrison was difficult.
Complicating the matter further, the French were blockading the Mexican coast, in support of Napoleon III's adventures in Mexico, and wanted to keep arms and supplies out of the hands of the Juaristas, while at the same time being lukewarm at best towards the U.S. A gun-runner heading into the mouth of the Rio Grande could have been shipping arms to the Confederacy, could have been shipping them to the Juaristas, or... could have simply sold them to the highest bidder regardless of affiliation. There's no safe bet on which direction those things would take after a sale in Brownsville/Matamoros.
Really the only things the Union had going for them in this situation were that Brownsville and the Rio Grande were nearly as isolated from the Confederacy as from the Union, due to distance and lack of effective internal communications, and that the river itself was shallow and could not support large cargoes in any case.