A survey of 500 decision makers at UK museums, galleries, theatres, stately homes and castles indicates a third of visitors have refused to follow social distancing rules or wear masks.
Galleries surveyed by insurer Ecclesiastical seemed particularly affected by "challenging behaviour", with four in five the target of a crime since the first lockdown in March 2020. This is, however, a decrease on the 90% that experienced a crime before the pandemic.
Those which had experienced a crime during Covid-19 had suffered an average loss of more than £90,000.
Anti-social behaviour is likely to be a concern when many reopen next week, the survey suggests. About a quarter of galleries said they struggled with rude or aggressive behaviour from visitors over the past year, some of whom show up even if they have been unable to book tickets.
Two thirds said their organisation is more concerned about crime since Covid-19 and 94% had introduced new anti-crime measures such as cameras, alarms or security patrols.