Minnows have become so expensive that I've started paying attention to keeping extras healthy between fishing trips. I keep tropical fish aquariums and simply apply some of that experience. Four factors are important: oxygen; clean water; temperature and food.
OXYGEN is easy with a basic air pump, the kind with an air stone (bubbler) attached by a plastic tube. A small/cheap one will do for 3 or 4 dozen minnows in a 5 gallon bucket. Don't over crowd. A portable air pump works, but batteries get expensive. Weight down the air stone if it floats.
CLEAN WATER can come from the tap if you use a commercial aquarium solution to remove clorine. StressCoat is a common brand, but all are about the same. Most recommend 1 teaspoon per 5 gallons of new water. I don't measure; just squirt a few drops in before adding the tap water. Change 1/3 to 1/2 of the water every few days; more or less frequently according to how crowded and warm the bucket is.
TEMPERATURE can be from near freezing to the mid-70's, just don't let it go up or down quickly. You change only 1/3 to 1/2 of the water at a time in order too avoid quick temperature moves. Use the cold water tap and your hand to make sure new water is not a lot warmer or colder than the old. In the South, unheated garage temperature is fine except for our rare spells below freezing and our hottest days. Cellar temperature should be ok anywhere.
FOOD is the least important factor. If everything else is ok, minnows will live a long time with no added food. In fact, over feeding, which pollutes the water, is worse than not feeding at all. Add no more than a very little goldfish flake food once per day.
An air pump (with airstone), declorinator and goldfish food (all small sizes) should cost about the same as a couple of buckets of minnows and should last more than one season..