EU foreign ministers also demanded the release of prisoners and access across the country for aid workers, medics and the press.
The ministers agreed at a meeting in Brussels to add Assad and nine other officials to a list of people affected by travel bans and asset freezes.
The European Union had already imposed sanctions on the president's brother, four of his cousins and others in his inner circle.
Activists say five people were shot dead at funerals in Homs on Saturday.
They were mourning some of the anti-government protestors killed by security forces after Friday prayers in the city, in central Syria.
Monday's EU announcement of sanctions came after US President Obama imposed sanctions on Mr. Assad for human rights abuses - the first western leader to do so.