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 <a href="http://books.google.com/books?id=sXt-ROLLNHcC&printsec=frontcover&dq=How+to+prove+it&source=bl&ots=BnMps81l65&sig=_Fj005EBu2K_YgyZ_5TNIDaKFmI&hl=en&ei=TBYvTeOEMYqr8Aao1J3YCQ&sa=X&oi=book_result&ct=result&resnum=3&ved=0CDUQ6AEwAg#v=onepage&q&f=false">How to Prove It: A structured approach</a>, 
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.