My nails are brittle and break easily. Would applying a base coat of clear polish help make them stronger or healthier?

No it wouldnt unless the nail polish specifically says it strengthens nails because if you just use any random clear coat it doesnt let the nails 'breath' so they end up weaker than they were.  more

Not unless you use a "base coat" that is specifically designed to strengthen nails. Nutra Nail and Sally Hansen make numerous products for this purpose. I prefer OPI or China Glaze and OPI makes a really good product called Nail Envy that has 5 different formulations to target specific nail problems.  more