The club announced on Saturday evening that they had received complaints about the behaviour of a group of supporters at the match, which Dumbarton won 1-0.
Livingston said anyone identified as being party to such abuse would be dealt with in accordance with the club’s zero tolerance of racism policy.
A statement on the club’s official website said: “Livingston has received several complaints this evening regarding the behaviour of a small group of so-called supporters who attended today’s game in Dumbarton.
“We understand these individuals allegedly arrived at the game drunk, and illegally secreted bottles of alcohol into the stadium.
“It’s further alleged that one or more of these bottles was later smashed inside the stand, and one individual launched an assault upon another.
“We have also been told that this same group shouted racist abuse at Darren Cole, among a list of other expletives aimed at our club captain.
“We are now fully investigating this issue and shall advise further in due course.”
Livingston said it would take just “one rotten apple to taint the reputation of the club and all our decent supporters”.
The statement added: “We fully accept that football generates passion in the stands, and from time to time frustrations boil over into verbal comments.
“These are part of the game - but only within reason. We operate a zero tolerance approach to racist abuse, and those found to have indulged in such acts will be dealt with accordingly.
“Violence and other antisocial behaviour is similarly not tolerated, and we will cooperate fully with Police Scotland should this ever be required.”