I don't have any practical experience dying basswood, but it would be worthwhile to explore this -
Why are you interested in dying it? Basswood is, in general, a very plain pinkish-white wood with no interesting figure to the grain. I have never seen it without a solid color applied, in fact, in a guitar application. If you want to put a transparent color on it (perhaps as the backside to a figured top), you would probably be at least as well served by tinting lacquer and shooting it on with a Preval (or a real spraygun, but if memory serves, you don't have one).
Bagman