I think it depends on how you word the sentence. In "three years of experience..." the subject is "years" so the verb is plural: three years (of experience) enable me... If you have it as "The experience (of three years) then the subject is "experience" and the verb is singular: The experience of three years enables me... However, most native speakers would not bother with the distinction (if there is one) and would probably say "enables" as a matter of course.