Saving money is not prohibited in itself, but saving it without paying its zakah is prohibited in Islam.
Sheikh `Abdul-Bari Az-Zamzami, a member of the Moroccan Scholars’ Association, states: “Saving money is not prohibited in itself. Rather, saving it and not paying zakah is what is prohibited. This meaning is confirmed in the following Qur’anic verse: “They who hoard up gold and silver and spend it not in the way of Allah, unto them give tidings (O Muhammad) of a painful doom” (At-Tawbah: 34).
Also, the Prophet (peace and blessings be upon him) said, “If you pay zakah on your money which reached the nisab (minimum amount on which zakah is due), saving it will not be (haram) hoarding up” (Abu Dawud).”