Mosques are not in the ownership of any person, rather they belong to Allah. No one may stop a person from entering the mosque; Allah says in the Quran:
"And [He revealed] that the masjids are for Allah" [72:18; http://quran.com/72]
However, if it is conceived that people will bring harm to the mosque or Muslims, then such people should be prevented because there is greater harm in allowing them to enter.
Aside from that, no one may be prevented from entering the mosque.
Anyone and everyone is welcome to come, and pray, if they wish.
However, it is a fundamental aspect of Islamic belief that no act of worship is rewarded unless it is done in the state of Iman (having faith in Allah and his Prophet Muhammad (peace and blessings be upon him)).