The Ghanaian, 29, set up the opener in a 4-2 second-leg defeat but the MLS team won the semi-final on away goals.
Oduro tweeted: “So sad that in this generation some fans still make monkey noises when black players are playing the game.
“Shame to all #alajuelense fans.”
He later added: “Say no to racism on the field.”
Oduro could play in Costa Rica again in the two-legged final as Herediano hold a 3-0 advantage over Mexican side Club America ahead of the second leg of their last-four tie.