Just recently I purchased a Realist pickup to put on my acoustic bass so I can record sound. I run the pickup through a cheap, $30 amp (an early model Crate GX-65) and then into Audacity for a final result. When I hit play on my recording, all I hear is white noise and a small hint of what might be my bass in the background. Is there any way to get rid of the white noise and have a loud, clear recording?
From what I'm getting, you're connecting your Crate amp directly to your computer? As in 1/4" to mini jack or something? There is no external audio interface (like this) in play here, right? And the pick-up works correctly with your amp (as topo morto asked)?
It seems to me that this is more an issue of getting the right gear. At first glance (I looked into the manual of a Crate GX-65), your amplifier does not have an output which is actually meant for recording. There's an EXT. SPEAKER output which is meant for the addition of an external speaker and there's the Insert in/output which is used to connect an external effects device to your amp. The use of either of these outputs for recording purposes is not the right way to go. However, you might use the external speaker, as explained here, but that's not a straightforward set-up.
Having an incorrect set-up will probably result in bad audio quality because your specific amp is simply not 'meant' to function as a recording device. Additionally - since your amp is not an audio interface - the audio will probably be recorded via the sound card of your computer, which, again, is not that suited for recordings.
So, to conclude: the problem at hand is an incorrect set-up and the solution/answer would for example be to get your hands on an external USB audio interface (Line6, Focusrite, ...) so that you can just connect your bass pick-up directly to the interface and have a 'proper' audio recording of what you're playing. In this scenario, you wouldn't even need your Crate amp (*).
(*) which is only for the best, because in your current situation, instead of the dry sound of the bass, you record the preamp sound of your amp - which probably is not that great