At the end of The Bourne Identity we see Bourne find Marie somewhere in Greece, even though she is in hiding. How does he find her? If he can find her so can other agencies. Why would he do it if he knows he will put her in danger again?
Well clearly he's in love with her and that is what provides his motivation for finding her.
Regarding the danger aspect, well at the end of The Bourne Identity he has taken out every operative the Treadstone program has thrown at him with pretty extreme prejudice and in addition to the significant losses he has inflicted to their operation they no longer have any active leads on him so it's not unreasonable for him to assume that if he's sensible and keeps a low profile (and he's highly trained at doing just that) that he'll be able to stay ahead of them. Certainly he's better equipped to do so than Marie is on her own and given that they already know about her and have her picture etc and know that regardless of whether she is physically with Bourne she is a lead/leverage worth pursuing she's actually probably safer with him then without.
Spoliers for The Bourne Supremacy :
And as we see at the start of the second film this approach works well for 2 years, it's only when Kirill (who is very nearly Bourne's equal in skill) with his access to Russian FSB resources and the likely Arbitrarily Large Bank Account of a Russian Oligarch gets involved that Bourne's safeguards fail.
He's a highly trained former spy/assassin. Tracking down his "target" is likely to be something they covered in that training. Marie is a civilian who has only minimal instruction in how to hide so for him to be able to track her down in ~6 months isn't a massive leap.