A great way to get a great recorded guitar or bass amplifier tone without using a microphone or making noise is to run the speaker output of the amplifier into a load box of some sort that can convert the speaker level signal to line level, which can then be recorded. (BE careful, some DI boxes can accept speaker-level signals, but still require that a speaker or load be connected to the amp for safe operation - double-check before you try!) The problem is that eliminating the speaker from the system means you've also eliminated all of the desirable coloration that the speaker provides - the raw amp output is going to sound pretty fizzy and anemic.

For a cure, try this: run the amp signal you've recorded through the speaker modeling portion of your favorite amp simulation software. Don't any of the amp or effects emulations - you've already got the amped sound, you just want the coloration of a speaker applied to the signal.

CAUTION: Don't connect an amplifier's speaker output to anything but a speaker or an approved load box/level converter - damage will result if you plug a speaker out into a mic-level or line-level input! Be sure before you proceed!