If by public, you mean government, and if by space exploration, you mean manned space missions, then no. As far as actual useful science, then for the foreseeable future we get more bang from the buck from unmanned missions. While it would be nice to put a self-sustaining colony on Mars, I don't see the political will existing to do that. (Especially not in the most cost effective way as it would require accepting a Mars with a radically different social structure.)
I don't agree with your pessimism when it comes to space exploration, but I understand your concerns. Personally, I think we should be using both manned and unmanned missions. I don't think political will should inhibit exploration. It was President Kennedy in 1961 that proposed we set foot on the moon by the end of the decade. We did so in July 1969. If President Obama or our next President proposed a manned mission to Mars within 10 years, I have no doubt whatsoever that we could do it. (To be honest, I don't think we should send people for the sake of doing so. If and when we do it, we should be establishing a permanent settlement.) Ideally, though, I think it'd be best that a permanent Mars settlement be a multinational effort among the great powers (i.e. US, UK, France, Russia, China, etc).
Kennedy made his pledge to put a man on the moon in an entirely military context. The Cold War was at its height: in 1957 the USSR scared the USA stiff when Sputnik beeped overhead every 90 minutes and in 1960 the USSR shot down our high altitude U2 spy plane and paraded the pilot and aircraft to show that the US was systematically invading the airspace of the USSR for espionage. Like the public support for a military response to 9/11, the public supported the space race as part of an attempt to recover military dominance atop the atmosphere.
Without a similarly compelling reason, I don't see how any president gets the public behind an all-out manned mission. I think a modest increase is the best that can be sold to the public.
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