Does a more durable base help a budget mattress last?

Yes. Even an affordable mattress lasts longer on a sturdy, even foundation that prevents sagging and supports the whole surface. A weak or widely slatted base lets the mattress flex and wear faster, and can void some warranties. Use a solid platform or correctly spaced slats, and rotate the mattress when the maker allows. Good support underneath is one of the cheapest ways to extend a mattress's life.