I taught the `transitions' course at a large state university a number of years ago, with reasonable success. The clientele of this (purely elective) course was mainly B students in calculus who would likely have done poorly in real analysis or abstract algebra, and would have had difficulty completing a math major.
To maximize the impact on students' ability to understand and produce proofs, several things were important:
a) The text was Velleman's How to Prove It: A structured approach, which is readable by average students, clearly delineates the structure and construction of typical proofs, and is full of problems which are elementary but not boring. (For a regular beginning analysis class I just ask the students to read this book---esp. chapter 3 as mentioned by Jon Bannon---and I discuss the basics of this material for a few lectures.)
b) The format of most class sessions was discussion not lecture. To have these students passively listen, like in their previous courses which they demonstrably failed to master, would be useless. Discussion was structured like in a humanities or language course, led by the instructor with specific goals in mind and calling on individual students to involve everyone and make sure they get it. The 22 students were informed that it was essential that they come to class prepared, having read the day's material and having worked the relevant problems, laid out in each week's syllabus.
c) Why we insist on ``proof beyond unreasonable doubt'' was explained, referring to the great discoveries of 19th and early 20th century analysis (especially regarding infinite sets and fractals) that demand the enormously skeptical approach to establishing truth which now dominates much of modern mathematics.
Many of the students were weak at the start and apparently benefited from all this. For one, this course was a big step in eventually changing his career from fisherman to gaining a Masters and working in a scientific software company. Another later did A work in a senior-level ODE course I taught. But I did not conduct a randomized controlled study.