Yes there is a brake. It is part of the transmission and brake assembly however.
You can open the front of this quite easily with a putty knife or any similar tool. Here's a youtube vid of it. Ignore the part about removing the console you do not need to do that 1st. The front panel comes off without removing the lid.

https://www.youtube.com/watch?v=BJBdwhGqL4Y

The transmission spins so don't get your fingers or hands into its area if the machine is spinning or about to spin. Start it in drain and spin, once spinning open the lid, this should stop it. The motor and belt pulleys will stop. If the trans continues to turn and coasts to a stop that's not right. You need a new transmission.

At this age it really depends on your willingness to do the work as cost to repair is high enough to push many into replacing the machine. I'm generally all for repairing due to the low quality of new machines. It's not particularly hard but I've done 100's so I may have forgotten my 1st. The only part a non-pro would find hard is the hub nut as it is far larger than most tool kits will handle. The rest is just following directions.

Note: There's always some suds overflow in the base plan so water there or evidence of it isn't a good thing to base a decision on. If water is actively coming down the main central shaft as it is running that is bad.