Dabo stepped up to take his side's sixth penalty of the shootout following a 1-1 draw after extra time and blazed his spot kick over the bar to hand the victory to Luton, who claimed promotion to the Premier League at Coventry's expense.
"The messages sent to Fankaty were disgusting and there is no place for any person who sent such vile abuse at any of our games, and social media is not a hiding place to make these comments," Coventry's owner Doug King said on Tuesday.
"The club and our partners would support Fankaty, and we will not hesitate in taking action against those people found to have done this.
"Coventry is a tolerant and multi-cultural city that we are proud to represent, and there is no place any form of racism or discrimination in person or on social media."
Coventry said they will work with police, the English Football League and the Professional Footballers' Association to ban and prosecute individuals responsible for racist messages.
