Nowadays (2016) Apple recommends more and more to use the URL related API of NSURL, NSFileManager etc.
To get the documents directory in iOS and Swift 2 use
let documentDirectoryURL = try! NSFileManager.defaultManager().URLForDirectory(.DocumentDirectory, 
                                 inDomain: .UserDomainMask, 
                        appropriateForURL: nil, 
                                   create: true)
The try! is safe in this case because this standard directory is guaranteed to exist.
Then append the appropriate path component for example an sqlite file
let databaseURL = documentDirectoryURL.URLByAppendingPathComponent("MyDataBase.sqlite")
Now check if the file exists with checkResourceIsReachableAndReturnError of NSURL.
let fileExists = databaseURL.checkResourceIsReachableAndReturnError(nil)
If you need the error pass the NSError pointer to the parameter. 
var error : NSError?
let fileExists = databaseURL.checkResourceIsReachableAndReturnError(&error)
if !fileExists { print(error) }
Swift 3+:
let documentDirectoryURL = try! FileManager.default.url(for: .documentDirectory, 
                                in: .userDomainMask, 
                    appropriateFor: nil, 
                            create: true)
let databaseURL = documentDirectoryURL.appendingPathComponent("MyDataBase.sqlite")
checkResourceIsReachable is marked as can throw
do {
    let fileExists = try databaseURL.checkResourceIsReachable()
    // handle the boolean result
} catch let error as NSError {
    print(error)
}
To consider only the boolean return value and ignore the error use the nil-coalescing operator
let fileExists = (try? databaseURL.checkResourceIsReachable()) ?? false