Very little. Championships are a nice bonus, of course, but are very much a team effort, not the product of the efforts of one player. And a list of great players who never hoisted the trophy -- by either playing before such a thing existed, or by never getting the chance, or by being denied victory when on the stage -- is studded with well-known names.
Curiously, a LACK of championships seems to influence opinions, adversely, regarding various Yankees players over the decades. Players for no other team have to put up with that.