This is because if we live longer, we will have more time to do the things we need to do. The best scientists will be able to figure out the world's biggest problems, without having to worry about running out of time. Also, we won't ever have to worry about suffering from diseases. Lastly, we won't ever have to lose our loved ones and we will be with them forever.
Extending our longevity is a good thing, provided it can benefit all people. It seems to me that longevity and health are related, so some of the research that focuses on one should benefit the other as well. We know that certain types of diets are healthier than others i.e. the Mediterranean diet, and that correlates with living longer in general. Plus scientific advances focusing on longevity may result in other advances which help cure disease.
It depends on what it meant by longevity. If finding immortality or preventing the aging of cells are for question, then it's iffy because such labors are fruitless. However, if longevity means curing diseases and other bodily inflictions that inhibit a longer life in the people who would otherwise live longer, then yes.
Medical science should attempt to extend the longevity of humans because humans may feel more responsible for their actions if they know they will be living with them for many, many years. A majority of the people who refuse to take care of themselves, others around them, and the planet we live on, claim to do so because "life is too short." I believe people would worry more about Earth and the well-being of mankind if they felt that they had a long enough lifespan to justify doing so.
People were never meant to live forever. The fascination with immortality has been around a long time, but it is something that needs to cease. If the world becomes overpopulated where will everyone live, how will they eat? Once the natural life span is over, that should be it for us.