By "crater", I'm assuming you mean a meteorite impact crater.
The closest impact crater to Austin, Texas is near Marquez, Texas. That's not very close, about 130 miles.
You can read more about the Marquez crater online here:
http://www.unb.ca/passc/ImpactDatabase/images/marquez.htm
Unless you're a geologist, however, it isn't worth the trip. It's covered in farm/ranch land, and doesn't look like anything special. This is because it is over 58 million years old, and has had a LOT of time to erode. It's also about 8 miles across, and so you'd have to be in an airplane to even see the whole thing; but even if you were, you wouldn't see much there.
Further from Austin, near Odessa, Texas, there is a much more impressive looking impact crater (impressive because it is MUCH more recent - about 50 thousand years old - and smaller - about 550 feet across). You can definitely tell it's a crater when you look at it, and not just from the air. Of course, Odessa is about 413 miles from Austin, so it's more of a trip to go see it.
Here's more information on the Odessa crater, which also includes a good picture:
http://www.unb.ca/passc/ImpactDatabase/images/odessa.htm