Since you have to go back to New Jersey, I think driving is your best bet though it will take you a while to get there. You will hate the train trip back and then the drive to Middletown.
There are safe places near the stadium to park, but it will cost you $10 or $15. It could take you close to 2 hours depending on traffic, especially the GW bridge and the Deegan Expressway. It could end up not taking that long either. That said, taking either train option including the one recommended by the previous poster could easily wind up taking close to as long.
Driving: Leave Newark Airport and take the New Jersey Turnpike (I-95) north. It's a toll road so make sure you aim for the cash only lanes. Follow the Turnpike all the way to the George Washington Bridge (cash lanes here too). The toll for the Turnpike will be about $2 and the toll for the bridge is $6.
After crossing the bridge, take the Major Deegan Expressway (I-87) south, less than a mile after you cross. Follow the exit sign for Yankee Stadium (maybe another 2 miles) and you are there. Don't go too far from the vicinity of the stadium either to park or when walking because it is a bad neighborhood. The best thing about driving will be getting back to New Jersey. You also don't have to worry about how late you stay at the game.
Train Option 1:
Follow the instructions of the previous poster, except I have no idea how to get to Exchange Place (it's in Jersey City). If you do that, it's the 4 train to Yankee Stadium, not the 5. The drawbacks of this route are the potential traffic getting to Jersey City and the unknown factor of how to get there and the parking situation (where, how much will it cost). Large parts of Jersey City are sketchy as well, but I recall the area of Exchange Place as being pretty safe. The other drawback is that if you are running pretty late on your way back, the PATH could take forever (it starts running only once an hour late at night). The 4 train will take about 30 minutes. The trains will only cost you $7 total if you go this way.
Taking the PATH from Newark would be a very iffy proposition since you would have to park in a really bad neighborhood. If for some reason Exchange Place is out, Hoboken is probably your next best bet (traffic and not real easy to get to and parking will be expensive). I'm also not sure about the Hoboken schedule to WTC at night, so it would be best to take it to 34th St., take the N, R or W to Lexington/59t St and transfer to the 4 there.
Train Option 2:
Don't pick up your rental car at first and instead take the Airlink (monorail) to the New Jersey Transit Newark Airport station (cost is around $12). Take the train to New York Penn Station (NOT Newark Penn Station). It's the terminus of the train. Take the E subway line to the Lexington Ave. stop. From here you have two choices. You can either transfer at this station to the 6 train and take it to 59th Street (the first stop) and transfer to the 4 to go on to Yankee Stadium, or you can get out and walk the few blocks from 53rd to 59th along Lexington and catch the 4 there that way (I would walk and the transfer will still work so no additional subway charge).
You can go back this way and the monorail at Newark Airport will take you to the rental car lots and you can pick up your car then. This way you wouldn't have to pay to park somewhere, but the trains will cost you more anyway (total of around $28). I guess you would have to stow your luggage somewhere though and if you decided to do get the car and park, it's very expensive around the airport (at least $20 for a remote lot).
If it were me, I would drive. You'll be much happier getting to Middletown this way, though it could take a bit of time to get past the stadium traffic. By the way, if you drive, to get to Middletown, go back the way you came, but stay on the Turnpike south and then take the Garden State Parkway south.
