According to the National Electrical Code, yes it should be a 20 ampere branch circuit. This circuit should also not have any other outlets, other than the ones for "laundry equipment".
So keep the 15 ampere circuit for your general use receptacles, and run a new 20 ampere circuit for the "laundry equipment".
Also note that the receptacles in the laundry area, must be GFCI and AFCI protected (210.8(A)(10), 210.12(A)).
#National Electrical Code 2014 ##Chapter 2 Wiring and Protection ###Article 210 Branch Circuits. 210.11 Branch Circuits Required.
(C) Dwelling Units.
(2) Laundry Branch Circuits. In addition to the number of branch circuits required by other parts of this section, at least one additional 20-ampere branch circuit shall be provided to supply the laundry outlet(s) required by 210.52(F). This circuit shall have no other outlets.
210.52 Dwelling Unit Receptacle Outlets.
(F) Laundry Areas. In dwelling units, at least one receptacle outlet shall be installed in areas designated for the installation of laundry equipment.