The Berlin Defense is drawish if we reach the Berlin endgame. Yet White doesn't have to go into the endgame, and can instead play 4; d3. However, I hear that the Ruy Lopez is getting less popular in top level because of the Berlin defense. This implies that 4. d3 somehow is not as good compared to the mainlines, such as 3... a6 of the Ruy Lopez. What makes 4. d3 not as good as the Ruy Lopez's mainlines for White?
In the Ruy Lopez White usually wants to play with c3 and d4 to claim the center. If you play d3 instead then you lose a tempo if you still want to go for that plan.
Of course it is not the only possible plan but then the question becomes, if you answer 3...Nf6 with 4.d3, why would you not play the Italian instead where you might get a similar structure after for instance 3.Bc4 Bc5 4.c3 Nf6 5.d3, but arguably keep a bit more flexibility.
Still, 4.d3 against the Berlin is a fine way to play, it just doesn't have a clear benefit over other similar setups.
4.d3 The main alternative to the move 4.0-0, it avoids Berlin endgame. White do not seek to squeeze the maximum from the debut, carrying the weight fight in the middle of the game. although elite grandmasters have recently become addicted to this continuation, his theory is still developing (comment from the book practical guide to the Berlin defense. Louis Bernard).