Exactly how bad is using 'mod' to clamp reduce numbers to a given range?
The answer is that it depends on how much larger $n$ is than $m$, and also depends on the application. If $n$ is much larger than $m$ (say 64-128 bits longer) then you are fine, as pointed out by @Ricky Demer. However, otherwise, you will have bits that have a bias. 

If you are using this key in HMAC, then it doesn't matter. However, in general, it can matter and can matter a lot. Thus, I strongly recommend against doing this without making $n$ at least 64-128 bits longer than $m$. If you want to see a work where small biases are used to recover plaintext, then Analysing and Exploiting the Mantin Biases in RC4 is a good place to start.