Ya I had no choice but share ideologies.
Sadly, this is often the case, and why it's a good idea to focus heavily on culture and tourism even if you aren't going for culture victory - it denies them tourism boosters. The way cultural "defence" works is somewhat opaque - while your resistance to other cultures' tourism is based on your culture (not tourism) output, your ability to resist ideological pressure seems to be based on your own tourism output (since the preferred ideology of a civ is determined by the combined influence of the civs following that ideology) - so you actually need high tourism to counter rival ideologies, not just high culture.
I didn't have open borders but i was trading with them a lot. Perhaps I should have put an embargo on them so they couldn't send trade routes?
That's an idea - I hadn't considered that use of embargos.
What else could I have done
Well, you've already had one idea I hadn't! Unfortunately the culture system as now implemented is not very dynamic - if you have an opponent with high tourism, and you don't have key culture boosters of your own like Sistine Chapel or Alhambra, all you can really do to respond is use two temporary measures: propose and win the World's Fair to double your culture output, and so defence against influence, and use your Great Writers to create political treatises. And neither GW creation and winning the World's Fair are entirely within your control.
The only other factor I can think of is religion - make sure you aren't sharing a religion with that civ; if you are, and have a religion of your own, make that your majority religion instead. If you don't, steal someone's missionary and create a city you can use to create faith-spreaders of that religion. There should probably be a negative tourism modifier for differing religions, but sadly there isn't.
EDIT: Culture-boosting Congress resolutions may be helpful (if you have access to Natural Wonders you can try Natural Landmarks - however you'll need to control the voting to pass it since the AI has a curious tendency to avoid settling NWs and will always vote against Natural Landmarks), but beware of Cultural Heritage Sites. The AI always loves this, and it will boost your culture output a lot, but as worded the culture counts as being generated by the Wonders. This means that hotels and airports turn it into tourism. If CHS is in effect, try to repeal it; any civ with a high enough tourism output to be a cultural victory threat will have a lot of Wonders.