This is a little tricky if your trigger is bulkified (processing multiple contact records).  As you are retrieving all of the applications for all contacts, there's no way that I know of to limit the query to only return one per id.

Off the top of my head, I think something like the following may work:

trigger rcdtypechg on Contact (before update) {


set<Id> chngRcdtype=new set<Id>();
    
    for(Contact co : Trigger.New)
    {
        if(co.What_are_you_applying_to__c=='Interlochen Summer Arts Camp')
            chngRcdtype.add(co.Id);
            
    }

    // create a map of application keyed by contact id. 
    // Then extract all applications, oldest first
    // Then iterate and put the app in the map based on the contact id
    // Each subsequent put will overwrite the previous one,
    // leaving a map containing the most recent app for each contact
    
    List<Enrollment_Opportunity__c> app=[Select Id,RecordTypeId, LastModifiedDate, Enrollment_Opportunity__c.Applicant__c from Enrollment_Opportunity__c where Enrollment_Opportunity__c.Applicant__c IN:chngRcdtype order by LastModifiedDate asc];
    Map<Id, Enrollment_Opportunity__c> appMap=new Map<Id, Enrollment_Opportunity__c>();

    for(Enrollment_Opportunity__c ap:app)
    {
        appMap.put(Enrollment_Opportunity__c.Applicant__c, app);
    }

    for(Enrollment_Opportunity__c ap:appMap.values())
    {
        ap.RecordTypeId='012T00000004cfd';
    }
    
    update appMap.values();
    
}

Caveat - I haven't compiled this, but hopefully you get the idea.  There's no doubt other ways to do this too.