Open borders will not destroy culture because immigration brings more cultures into contact with other cultures. Freer movement creates more diversity and expands cultures by introducing new ideas and customs.
Borders do not preserve culture; people and culture create and preserve borders between cultural differences. Abolishing national or politically determined borders would not destroy culture because societies already distinguish themselves from one another by natural geography, language, or even sports teams. 
Culture transcends borders.
A person’s culture does not disappear when they migrate to a place with a different culture. People adapt their own cultures to new cultures. Borderlands, ambiguous spaces between politically determined borders, show how cultures can mix, form new things, and cause antagonism at the same time.