In Earth's deteriorating state, we will need to find new shelter before we just can't continue life on Earth anymore because of increasing or decreasing greenhouse gasses. If we lose or gain a certain amount of greenhouse gasses we're in some deep trouble. Greenhouse gasses are insulators, they essentially keep us balanced while protecting us from the Sun's harmful rays. If we lose the insulators then we are less protected leading to an unstable environment and dramatic climate change, if we gain more insulation then warmth is absorbed and can't be let out, leading to again an unstable environment and dramatic climate change. This is why we need to find another place where we can continue life even if it is dangerous. I think staying here on this slowly weakening planet, is more dangerous than to travel to places that may be habitable that we have found or keep searching.
Yes, I believe that we should send a manned ship to Mars. Furthermore, I think that we should also equip this ship with long-life sustainability features, so that we could observes any type of weather patterns and life forms that may be encountered by the ship. The manned ship would also be able to send information back to us which would help us learn more about Mars. Though our knowledge of Mars is vast, we know probably a tenth of 1% of all there is to know about Mars, and by sending a manned ship to Mars, we could improve on this knowledge. Someday, hopefully, we could even build a space station on the actual planet of Mars, which would be the ultimate tool in observation and discovery.
As of now, it is proposed that Mars is not habitable by human beings. However, how are we to know that Earth was not at one point uninhabitable? Our earth is deteriorating slowly and we may have need to move to another planet. While it is unlikely that there is another planet we can move to, it doesn't hurt to try.