Bosses at the heritage line reported an increase in incidents of people at various points of the 16-mile route, which runs between Bridgnorth and Kidderminster.
Helen Smith, general manager of SVR, said anyone who walks on or beside the line is breaking the law and putting themselves in danger, and that security along the tracks has been increased.
"Although there are currently no public services as a result of the lockdown measures under the coronavirus pandemic, we do still have unscheduled engineering trains moving around the railway as part of our care and maintenance operation," she said.
"We are under tremendous pressure at the moment and the last thing we need would be to have to deal with an injury or worse on the railway as a result of trespass.
"We're stepping up security along the line and urge members of the public not to use the railway as a footpath.
"Please come and enjoy the beautiful scenery from on board one of our trains once we're able to resume services."
Despite public services along the line coming to a halt, walking on or beside the route is illegal under section 16 of the Rail Regulation Act 1840 and could lead to prosecution and a fine of £1,000 or up to one month in prison.
The SVR closed its doors to visitors on March 15 as part of the lockdown measures following government advice.
Since then railway chiefs have raised concerns about its "uncertain future" and launched an emergency appeal for donations as a result.