Our moral responsibility in order of importance: 1.) Look out for number one (i.e., yourself) 2.) Don't harm anyone 3.) Help the most unfortunate 4.) Help anyone who needs help. (Sad to mention the first one. But it's still necessary in the world today.)
I don't know if it's a moral responsibility or not but I do feel the need to support others. I donate monthly to MSF. I am part of a micro loan scheme supporting small business in South America, Indian subcontinent and Africa. I support the guardian newspaper to help ensure it's independence. This is the first time I have mentioned this here. I normally keep these things to myself but since no-one I know uses this site.......
I think it depends on where you live and the culture you've been brought up in. I certainly think that in the US people are very self-centred and not at all understanding of the problems of those less fortunate than themselves. In other cultures it's normal for everyone to feel a social responsibility to help others, but it seems that the unbridled capitalist mindset of US citizens is; 'me, me, me, and devil take the hindmost'. A clear example of this is healthcare. In the US no-one is prepared to join in and pay for equal treatment for everyone, whatever their financial situation. In contrast, just about everywhere else in the world, people do share the costs of healthcare through mutual insurance, or in the case of the UK NHS, through general taxation of everyone. The benefit of the shared approach is that everyone has a health safety-net that benefits the whole of society, and the country is then less burdened with sick people dragging on the economy.