I am reading this paper about cyclers between Saturn and some of it's moons. In the beginning is discusses the Earth-Mars cycler as "suffering" from the "risky requirement to perform hyperbolic rendezvous." So, what is a hyperbolic rendezvous? Does it mean that the orbiting vehicle must enter a body's SOI at a high-speed hyperbolic trajectory? Is it the speed which makes it risky?
The sentence reads "Earth-Mars cycler applications generally suffer from long repeat periods, infrequent launch opportunities, and the risky requirement to perform hyperbolic rendezvous"
The bolded portions are important parts of that sentence.
If a rendezvous is botched during planetary fly by, there won't be another opportunity until next fly by. The taxi is now on a heliocentric orbit that won't fly by a cycler or planet for years. Passengers aboard are as good as dead.
The Saturnian moon cyclers also fly by the moons at hyperbolic trajectories with regard to the moons. But trip times and launch opportunities are on the order of days or weeks instead of years.
Here is a perhaps an over-simplified pic of two possible Mars cyclers:
As you can see fly bys are infrequent for both VISIT 2 and Aldrin Cyclers, though Aldrin flies by both earth and Mars more often. But the Aldrin Cycler has much bigger Vinfinity (also known as hyperbolic excess velocity). The Vinfinity vectors are colored red in the illustration above. You can see the Aldrin Cycler Vinfinity at Mars fly by is quite large, around 10 km/s
The Aldrin cycler also needs its line of apsides rotated about 50 degrees each synodic period. Buzz Aldrin hopes to accomplish that with earth's gravity instead of propellent.
Here's a spreadsheet that shows synodic period and trip times between moons of Saturn as well as moons of Jupiter. If you play with it you can see the tempo for travel between gas giant moons is a lot quicker than travel between planets.
Coming from one of the sources, a hyperbolic trajectory rendezvous is dangerous because if the rendezvous doesn't do as planned, the cycler taxi will escape the body's gravitational pull and venture out into space.
From the paper: "Apollo missions included a similar risk when the lunar module docked with the command/service module in lunar orbit. If this rendezvous failed, two of the three astronauts would not make it home."
The risk with hyperbolic rendezvous could be alleviated by having a "rescue cycler" following e.g. one day after the "station cycler" in a very similar orbit. The rescue cycler would consist of a chemical rocket engine with a generous fuel tank. If the crewed taxi misses the rendezvous, the rescue cycler would set off to rendezvous with the taxi instead, and tug it to the station cycler. This would give a second chance and larger safety margin, without increasing the fuel mass of the chemically propelled crewed taxi. The rescue cycler could be relatively economically put in place with a slow ion thruster and would only need to be refueled if it has been used.