The shortest answer to your question is yes. Seniors are more likely to have injuries, general weakness, or other extenuating health conditions due to their age or other activities when they were younger. More care has to be taken toward the beginning of their journey, but once they get started it will drastically improve their quality of life. That said, their bodies will still respond to training stimuli. Improvement will be slower and fatigue will be higher due to the age factor. However, they will improve, and they will get stronger/faster/more mentally quick just as younger folks do.
Step 1: Assess and Correct Base Fitness
If the senior is very much out of shape, has high blood pressure, history of heart disease, or have lost a lot of muscle due to sarcopenia, the steps at the beginning will be very modest. You may have to consult with a doctor to help design something they can do. Someone in their 60s is likely still strong enough to embark on fitness. However, if the person is in their 70s or higher when they start, there's a good chance they'll have some basic work to do.
Start slow. The first step is to get them moving more often. It's likely that stuff hurts and they don't like hurting. It takes time to figure out when they are wimping out or it truly is a chronic problem they have to work around. Sometimes the effort involved just in standing up alone takes all the vim and vigor they have. If that's the case, you may have to start them off in the weight room.
A leg press can be a an effective way to get them moving while they are seated. Start on the lightest setting, and over time add weight until they can leg press their body weight. That will help them tremendously with just standing up.
You will likely need to get them to do some sort of back exercise so they can keep their torso as erect as possible. If they already have severe kyphosis there may not be a lot they can do. If they haven't developed it yet, the back work will help them tremendously. Same approach with the leg press, use a cable row machine on the lightest setting and work up to about 50% of their body weight. Then graduate to back extensions. When they can do that bodyweight only movement, they will likely have the confidence they need to do most things.
Step 2: Find something they will want to continue to do
The older the trainee, the more brittle their bones and cartilage. That limits the types of things they can do. Sports that require a lot of agility may be out of reach, or highly unadvised.
I always suggest that there be some element of strength, mobility, and conditioning in your activities. Strength training provides a number of quality of life benefits as well as enabling the trainee to perform another sport better. Mobility also improves the quality of life as you can move better and with less pain. Finally, conditioning helps the cardiovascular system respond to the demands that the trainee now has.
As the trainee starts to feel better and more enabled, they are more likely to keep doing what they are doing. If they used to be completely untrained, they will start to gain enough muscle mass to support being active. They won't feel as frail.
At this point it's a question of finding something they can do that they enjoy. The more you enjoy something the more you will continue doing it. Think outside the box. Tai chi (or taiji) uses slow flowing movements that even older students can perform even after many years of abusing their bodies. Running is an option, and I've witnessed a few 70+ year old guys beating much younger guys in 5Ks. I also know a 79 year old power lifter who handed his cane to the judge at my last competition as he said "excuse me, I have to go squat". He was a man with a bad back and a minor case of permanent kyphosis, and was able to squat, bench press and deadlift (his final pull was over 315 lbs).
The biggest challenge is finding something that is either competitive or social (or maybe both) that they will want to continue doing. It may be that while you are working on the base level of fitness that they take to the weights. Great. It might be that now they can move better power walking and then running is actually enjoyable--esp. if they meet people they wouldn't otherwise.