Nate Silver has a post on the potential outcome of the debt-limit deal, in which he draws on public polling data from 2010:
The table below reflects the views of Democratic and Republican adults toward cuts in 18 areas of federal spending as derived from the 2010 General Social Survey. The scale runs from 0 (meaning that voters would like to see increased spending in that area) to 100 (meaning that voters would like to see spending cuts).

His post is mostly about defense spending, but note the category that is number three in terms of preferences for cuts — “space exploration.”
Four points.
First (the trivial one), it’s not a recent poll, but I’m not aware of anything that has happened in the past couple years that would change this, with the possible exception of the Shuttle retirement, and potential unhappiness about that, particularly given the nonsense and hyperbole that it represents “the end of US human spaceflight” (if not the end of all human spaceflight, US or otherwise). On the other hand, I don’t know the methodology.
Second, there is no weighting for the amount of spending. I haven’t seen the poll questions, but I’d be willing to bet that prior to being asked about their priorities, the respondents were not informed of the size of the thing they wanted to cut. For instance, there are no doubt many people who think that we spend as much on NASA and foreign aid as we do on defense, and if I thought that was the case, I’d want to slash them, too. The reality, of course, is that both NASA and foreign aid are a tiny fraction of the money that we spend on defense (as is appropriate). I think that when asking question like this, the polling should be done in a manner that would be reflective of how a rational decision maker would do the cutting, taking into account both the utility of the activity, and the effect of the cuts on the budget (that is, all other things being equal, a larger budget is worth expending more political capital to cut than a tiny one). Like Willie Sutton’s explanation as to why he robbed banks, we should go to the high-ticket items because that’s where the money is. Which means, of course, that entitlements should be first on the chopping block, whereas they are one of the lowest priorities for cutting according to the polling.
Third, in addition to being one of the top three (at least among Dems, though it’s high among Republicans, too) it was one of only two items on which there was a majority of two parties in favor of cutting (the other was foreign aid, with even higher numbers). Defense was favored for cuts by the Dems, but not by Republicans. (As a side note, Republicans don’t seem to be interested in spending cuts in general — there are very few categories that got majority support from them. This is a partial explanation for the rise of the Tea Party.)
Fourth, like all such polls, it is flawed in how the question is framed, and that is particularly the case with the space question. Without taking the time to dig into it, I am assuming that the respondents were simply presented with that list as worded, and asked if they favored cutting the item. Such a poll will only give impressionistic results and, like the issue of how much is spent in each category, is highly dependent on the individual’s interpretation of what the words mean.
I could write a long essay on this (and I actually am, as a chapter for a book), but “space exploration” is such a nebulous phrase as to be meaningless for making public-policy decisions. It’s just short hand for whatever NASA is doing, most of which has nothing to do with “exploration,” nor should it if one reads the agency’s charter. I wonder what the responses would be if instead of whether or not they were being asked to support space “exploration,” they were asked to support space “development,” or space “technology,” or space “industry”? And told how much we are actually spending on those things, with a pie chart compared to the others?