I've only done a little hobbyist-type recording so a more experienced person might give better suggestions, but here are a couple of thoughts:

1. use the best-quality microphones you can beg, borrow, or steal, that you can make work with your system. (there are adaptors if you have a plug of one size & a socket of a different size). When you say "tape deck", I don't know if it's a multi-track recording deck or a standard cassette player with one or two jacks for microphones.

2. If possible, use at least 2 mics

3. Putting the mics close to the instrument will give you the strongest signal to override the noise of the tape. However that means the mic will only pick up sounds from one part of the instrument, and a mountain dulcimer emits sound from a large area. That's why multiple mics are better.

4. Play around with mic placement and angles to get the best possible sound. As a starting point, try one mic aimed at where the strings are being strummed, and another aimed about halfway along the soundbox, about a foot away, both mics angled about 45 degrees from perpendicular to the soundboard. Play with angles as well as distance - most mics are somewhat directional and you can use this to emphasize or reduce certain sounds.

5. If you're stuck with only one mic, you might need to back it farther away to capture the sound from the whole instrument. Maybe 4-6 feet? But then you'll pick up all the other noises in the room, and the signal going onto the tape might not be strong enough.

Good luck! Getting a faithful recording of a quiet acoustic instrument is quite a challenge.