We have enough military bases in the area that they couldn't do anything, and honestly I don't think they'd have enough troops to staff it significantly. (they have only 37.5k) Considering what they'd have to invest to make any kind of significant base, I'm pretty sure that this is just a political ploy because they think we'll say no - they probably want something else. We should say yes just to see what they do.
For a country that size, Ecuador has a very large military as a byproduct of their repeated wars with Peru. While it is a political ploy, it isn't total nonsense.