Well - to be quite frank the type of problems you mention is more likely to
not be present in younger, healthy, well trained individuals - so for a
start - don't grow old and  frail, don't gain excessive weight, don't stop
exercising - and don't get sick.

So for the remaining part of us that did grow older, did gain weight and
eventually did get sick -
If you are aware that blood is most easily drawn from for instance your left
arm, tell them.
If you know that a certain vein is good for sampling - tell them.
Do not tell them that it is impossible, that the last time blood was drawn
they had to try for 1 hour and that you have a law suit running due to
unnnecessary suffering from the last vein puncture

If you could try to control your anxiety - do it! Nervous patients,
uncomfortable patients or scared patients are most likely to experience
problems. - And please note - I am not saying the phlebotomist is not
aproblem, but very often the problem lies with the patient. Also - if you
are known as "the trouble patient" - the one that always complains, shouts
or scream - the one who is in pain even before unwrapping the needle - this
could heavily influence the situation.
The lab people is actually doing their best - they want this process to run
as smoothly and quickly as possible - they are well trained and highly
routined in this job.