Grab a slightly damp paper towel and clean it up.

If you are really concerned, tie a scarf, hankie, something you can wash/sanitize if you don't have a disposable dust mask for over your mouth and nose.

Wear disposable gloves
Spray the urine and droppings w/mixture of bleach and water and let sit a few minutes. The mix of sanitizing bleach solution is approx 1 part bleach to 10 parts water.
Use a paper towel to pick up the urine and droppings, and dispose of in the garbage.
After the mouse droppings and urine have been removed, disinfect any items that may have been in contact.

When that's done, clean and disinfect the whole area -
Mop floors and clean countertops, sink, stove, refrigerator top with the same strength bleach solution.

If they are only under the sink, are there openings aroung the fittings or pipes where a mouse could get through...it can be much smaller than you might imagine. If you see any possibilities, take some steel wool and a screwdriver and pack the gaps tightly.

My BIL was having an awful time with mice in their lovely completely renovated farm house. Open those gorgeous cherry cabinets and you could have been seeing an ad for Tupperware, no food was left in the original packaging, all was protected. As a last resort, he pulled out the built in dishwasher and found the tiniest of openings around the plumbing there, problem solved. Traps with peanut butter will be your friend.

The above two responses are spot on.

This is just my personal opinion, but if you are a reasonably healthy person, not immune suppressed or undergoing treatment for a major illness or injury, then do what rrah suggested - pick up the droppings with a wet paper towel. I have vacuumed and swept up mouse droppings over the years with no ill effects.

Then start working on keeping them out with the suggestions morz8 gave you.

Thanks, everyone, for the advice.
I'm scared to death of mice and have been putting this off the last few days, lest I see one while cleaning up his gross little mess, but mice generally come out at night, right?

I did indeed see a hole or two in the way back - they look too cleanly cut to be mouse holes, but I bet that's how the little buggars got in. I've heard spray foam works well to block openings up, and I think I might call up the landlord to ask permission to hammer some metal over them. Gosh I don't want to do this, haha.

Unless you buy a pest block spray foam, the type that has a deterring bitter noxious taste built into it, mice will chew through it....The typical insulating sealant types aren't appropriate.

Some of our renters stored open 20 lb bags of rice under their sink and roaches crawled all over it. I would never store food connected item under the sink. I would stack them on the counter top if there was no other storage.

Best storage - glass jars. Mason jars are easy to get in many different sizes (even for free on freecycle...)
There are plastic tops available for regular & wide mouth.
No insect or pest invasion...

Go to Home Depot and buy a $4 bag of steel wool, probably located around the paint supply area. Not steel wool pads, just steel wool. Pull off tufts of it and stuff in the holes.

I like to use a little spray foam to help hold the steel wool in place. As previously described, stuff the steel wool in the holes with your finger, screw driver, or even the handle of a wooden spoon. The stick the straw of the foam in there and give it a little blast. The foam sticks to everything and helps the steel wool stay in place. It also give you a visual indication if anything is trying to chew it's way through.

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The foam will also stick to your hands so wear disposable gloves.

I have to say, Mike Kaiser, I like your solution the best! I'm having a problem with squirrels getting into my attic. I replaced all the vents under the soffett in the last couple of years and you can already see where those guys have managed to get through the screen in the vent; they've even bent the louvers. I will try steel wool and hold it to the vent with some of that spray foam. Thanks for the idea!

All the responses have been spot on so far. Cleaning and sealing the place down is a must especially since you use it for storing food. I also like the idea of getting jars or other kinds of containers for additional protection. It would be kind of a hassle to put the products into containers but it is definitely worth it. I would be careful of what chemicals you'd use to clean the cabinet though. It might damage the wood and make it easier for pests to chew through.