How does the transactions interact if you set a different isolation level per connection?
Every connection will work with its own transaction isolation level. Setting SERIALIZABLE level for one connection doesn't change isolation level of other connections (new or existing).

However interactions between transactions performed on different isolation level are quite complicated. Everything depends on many factors:


If different transactions operate on the same table/tables?
How your database engine implements transaction isolation levels (pessimistics versus optimistics locking)?
What is characteristics of access from your app to particular tables?


If you have to work with SERIALIZABLE level for some connections in your app you should know answers for above questions. 

For example. If you have database with pessimistics locking (e.g. IBM DB2) and you have long running transactions with SERIALIZABLE level it is very probable that your system would hang (or slow down).