Romania and Bulgaria are in the EU.
So yes, the EU borders would either have to protected very hawkishly, or there would have to be kind of a visa, that allows for travel in the EU, but no access to the basic income.
It is always a bit obscene to provide strong solidarity with e.g. Romania, but not with e.g. the Ukraine. But in the end, if the EU wants to be more than a free trade zone, it will be very difficult not to have somewhere are rather sharp border, either geographically or ideationally with giving EU country citizens largely more rights than non-citizens.
The alternative is less solidarity at at all, or solidarity with the whole world, which is not practically to the degree we want to have within our own society.

Der Amerikaner ist die Orchidee unter den Menschen
Volker Pispers