I read this post which deals with openly paid internships, but I don't think it fully answers this specific question:
When looking for a paid internship only, is it a good idea to include this in your reusme cover letter (in a non-chalant way) such as "I am seeking a paid inernship..." as opposed to "I am seeking an internship..."? The answer should apply to both soliciting you resume (sending it out to companies who have not placed a hiring advertisemnt) and also to those who have advertised but not specified whether it is paid or not. I'm not sure this is the appropriate place, or wait until the interview (if you get one).
I can think of a reaons for both: Specifying that you are looking only for a paid internship may turn off potentional employers who have not requested a resume (in the first place) from first considering you, but could save you (and them) the cost and time of inteviewing for no purpose.