Bernadette Smyth (51), from Suffolk Street, Ballymena, Co Antrim was convicted of two counts of harassment between January and February this year.
Speaking on her behalf outside court, solicitor Aiden Carlin said they would appeal.
According to the Irish Times, District Judge Chris Holmes told Belfast Magistrates' Court he was in no doubt Smyth's conduct outside the city centre clinic had been threatening and should not be considered a minor offence.
He said: "I do not feel it is acceptable for anyone to be stopped outside this clinic in any shape, form or fashion and asked about their identification."
The judge warned that Smyth, who heads the Precious Life pro-life organisation, could face jail or community service.
He said he would also be imposing a restraining order barring Smyth from an area close to the Marie Stopes clinic.
Judge Holmes said he believed Smyth's "entire purpose" was to stop the work of Marie Stopes. He added: "This lady is someone driven by very strong views and feelings."