I've used the Foursquare API for a lot of different kinds of applications, both web and mobile. In my opinion you should not store any venue information in your application for a couple of reasons.

First, I think the terms and conditions you agreed to when signing up for an API key prohibit you from doing so. Second, That data is subject to change at any moment so you're better off just referring to them to get it whenever you need it.

If you have any other questions about the Foursquare API I'd be glad to talk to you and see if I can help.

Thanks

According to experience of our team we have stored venue_ids and check-ins names in our own database to be able to reflect these on a map or open 4sq app with a corresponding venue.

If you need to, you can do some caching into your own database, that could help instead of accessing the foursquare API each time you need information. Depending on how you DB/application is structured, this is particularly useful if you need to run jobs or need to access the foursquare data in a static way. If you're using a NoSQL database, I encourage you to dump the full venue information in your stored document. If you're using SQL based, store it as a Blob and reference to venue_id.

Does it help?