In this note we consider the question of whether there are infinitely many primes in the intersection of two or more Beatty sequences ⌊ ξ_jn + η_j⌋, n ∈ N, j = 1,...,k. We begin with a straightforward sufficient condition for a set of Beatty sequences to contain infinitely many primes in their intersection. We then consider two sequences when one ξ_j is rational. However, the main result we establish concerns the intersection of two Beatty sequences with irrational ξ_j. We show that, subject to a natural "compatibility" condition, if the intersection contains more than one element, then it contains infinitely many primes. Finally, we supply a definitive answer when the compatibility condition fails.