Star Trek isn't possible, but yes, if NASA got more funding, it could certainly do much more.
In my opinion, humans should have a globally unified space agency rather than individual ones for each country/private organisation. This will make things much more effective as we would be pursuing a common goal, rather than competing with one another. It would also mean funding could be more spread out, and this organisation would have a way better resource pool and more financial resources than NASA on it's own would.
However, one must consider, why are we looking to the stars when we still have much to do on Earth?