It will effect cost - the more bidders the more it is going to cost to get to the top - although it is not a straight relationship, i.e. the bidding can also be effected by expected returns of the bidder or Google's valuation of the keyword, such that some niches with few bidders can be much more expensive than other markets with hundreds of bidders. It has no effect on quality score.
Popularity of keywords does make them more expensive but you can bring down this cost. This can be done with the right choice of keywords along with the right choice of either broad match, phrase match, exact match as well as having effective ad text which uses highly relevant content to the search term. Quality score is not based on the popularity of keywords.