Try flame tests (as Albert said) Barium has a green color and strontium a brick-red flame.
The best way to distinguish the two cations, however, is trying to precipitate the Ba2+ with potassium chromate in acetic acid acid-sodium acetate buffer. Yellow precipitate shows Ba2+.
If Barium is present filter the chromate, neutralize with Na2CO3 and test for strontium with sulfate. White precipitate of SrSO4 shows Sr
PS: Use a spectrometer to distinguish the colors.