I hear this all the time. A player gets older and what I hear is "I am getting slower in my reactions." So, what do you do? You might never get a lot faster, but you can get a lot smarter.
After you watch the ball come into your racket and the ball goes to your opponent's side of the court, quit watching the ball. Start gaining some information that may help you. Remember that you can not make line calls on your opponent's side of the net. Instead, make mental notes to check your opponents position, is he/she offensive (attacking) or is he/she defensive (in trouble or neutral, i.e. rally)?
For example, if your opponent is offensive, get ready to be attacked. If your opponent is defensive, get ready to attack. And if your opponent is in a neutral position, look to rally and continue the point. Also, look at your opponent's racket preparation. If his or her racket is way back, look for a drive. If his or her racket head drops down, look for a lob and if his or her racket sets up high, look for a chip or drop shot.
Lastly, look for your opponent's general position to the ball. If your opponent is close to the ball, look for a cross-court return. If your opponent is away and reaching for the ball, look for an up-the-line shot.
None of this information will come to you if you just watch your ball go over to your opponent's side of the courts.
Good luck on the courts!