properly design the slope.

f-d

¡papá gordo ain't no madre flaca!

most important would be flat enough slope and secondly, if you can zone the embankment and put the granular material in the shell and clay in the core that will increase the stability.

My apologies for not framing the question correctly.  Let's assume you are stuck with what was built.  A 2:1 slope with exposed clay, no shell, no rip rap, nothing.  This is a problem for an engineer dealing with a 60-year old earth dam.

Water reservoir or hydro dam?  I don't see how you will be able to do much (fully in the wet) to the existing submerged slope.  I am presuming that you have done site investigations and that your stability analyses show that you have a marginally stable to unstable situation.

Presuming this is the case, you might consider stabilizing berms on the water side.  These can be built by carefully dumping on the water side in a controlled fashion. You could also, in conjunction, build a riprap face on the water side. - not easy but nothing with 60 year old would be easy . . .  Run your stability computations with such geometric changes and see how your stability is increased.

Years ago, I was involved with the stabilizing of a causeway in northern Quebec on very soft (Su about 10 kPa) clay.  Luckily, it was winter and the lake had several feet of ice.  We were able to lay a stabilizing berm on the ice - it deflected and when melted, laid the stabilizing berm on the lake bed.

well the other obvious way (if this is a storage reservoir and not just a dry flood control dam) is to control the drawdown so it is not considered "rapid".