I thought this was somewhat common knowledge, but the AP decided they needed a poll to determine that this was in fact the case. In said poll, 70 percent of those asked said they opposed a reinstatement of the draft, with nearly half of all pollers being strongly opposed.
Some of the interesting things: Women were more likely to be opposed than men, people over 50 were more likely to favor the draft than younger people, and Republicans were more likely to favor the draft than Democrats. So, a 50 year old republican male is more likely to support a draft that would take people under the age of 28. Don't read so much into that, though. Many of those people were draftees themselves, and they view it as a one's duty as an American citizen to serve their country, and thus would be more likely to support a draft.
I would bet, if you polled military members, there'd be a stronger support for a draft than against, for similar reasons.
The problem is that we are currently in a struggle that is costing American lives and has no visible end, and as it becomes more and more unpopular, fewer and fewer Americans will volunteer for service. This will strain an already stretched military infrastructure even more than it already is.
I can't tell you whether I would support a draft or not. I volunteered to serve, and I think it's important for people to give to their country, but I'm not sure I subscribe to the idea of conscription again. I'm afraid that's the best personal answer I can forward. All opinions are welcome, perhaps I can lean more towards one side of the fence...
As far as the strain on the military... Our President has said we won't withdraw anytime soon, so the relief from that end isn't coming. I suspect that recruitment woes are going to continue as this drags on with no conclusion in sight. Perhaps the only way to ensure a properly functioning military is to reinstate a draft, for the sake of American lives and the continued American way of life...?