I have/had a pretty sever needle phobia and I have been able to manage with keeping up w/ and taking the copaxone.
There is a bit of a sting after you inject but I find that for me I can minimize this by warming the area. I also get redness and some hardness. But it's well worth it if this shit keeps my brain intact.
I would say go for it, the needles suck (mostly cause I hate them) but progressing is worse than needles.