Officers have also fined 48 people for failing to wear face masks or travelling illegally during lockdown.
One was arrested and reports on two others were sent to the procurator fiscal for prosecution.
The Covid-related crimes included 15 assaults.
BTP said in some of these, suspects claimed to victims they were infected whilst committing a crime.
There were also two cases of hate crime where victims were told by their abusers that they or their countries of origin were to blame for the virus.
Three rail workers were among those spat at, with most of the incidents in Glasgow.
BTP said many of the cases were still going through the judicial system “so it would be inappropriate to comment further at this stage”.
Chief Superintendent Eddie Wylie, BTP’s commander for Scotland, told the Scottish railway policing committee: “BTP, in the Covid regulations, have recorded 51 breaches and another 55 crimes associated directly to Covid.”
A BTP spokesperson said: “Under the Health Protection (Coronavirus) (Restrictions) (Scotland) Regulations 2020, the [Scotland] division has issued a total of 51 fixed penalty notices to date to individuals who were away from their home address without a reasonable excuse, and for non-compliance with face covering requirements.
“Our officers continue to patrol the rail network, supporting rail staff in engaging with passengers about the importance of maintaining social distancing and reminding those who do need to travel to wear face coverings.
“We are confident the public recognises the need to act responsibly and play their part in helping to protect each other by following the guidance and restrictions set out by the [Scottish] Government, and by complying with the requirement to wear face coverings.
“Anyone who fails to wear a face covering without an exemption is being encouraged and supported to do so but, in some cases, where appropriate they can be refused access to the rail network or issued with a fixed penalty notice.
“Our officers are there to properly challenge those who are not complying with the health requirements and to provide visible reassurance to all those using trains and services.”