|Any chance your machine has a delayed-start function? I love this on mine for just such occasions, so you can do the work at night, but you have freshly-laundered clothes to hang out first thing in the morning, without their having to sit for long.
If not, personally, I'd probably prefer not to have clothes sitting for too long, so would likely go with the morning option. But if putting things in at night is much easier, it's certainly worth trying it. You'll know by smell if they've gone musty, you'll deal with it, and probably not leave them overnight again. If they don't smell, then all is good. It's probably something that really depends on climate, washer, etc., so I'd think it'd be worth trying if it makes your life easier. I wouldn't think that the mustiness would be something that would "build-up" over time, so I would think that as long as they smell fine, you should do whatever works best for you.