Every individual who has Down Syndrome is impacted in various ways.
One of the most important things to remember: either way, that individual is still a person.
Persons who have Down Syndrome are still students, brothers, sisters, children, aunts, uncles, etc, the list goes on. Not everyone who has DS live the same lives, and the symptoms vary from person to person.
Even as someone who's been exposed to the DS community my entire life, I still catch myself with how to correctly and appropriately describe Down Syndrome. I believe your connotation is incredibly important, as the way you describe the community can have an impact on those who have no exposure to it.
Rather than saying "people with Down Syndrome," or "people who suffer with Down Syndrome," use "Persons who have Down Syndrome." This way, you remove the group phrasing, which implies that those who have DS are different from us, and are all grouped as the same. By using persons, it maintains the individuality of each person who has DS. (I myself am still working on this!)
Remember, the wording that you use reflects your feelings for that individual, so stay away from words with negative connotations, like suffer. Those who have Down Syndrome are not suffering, nor are they victims. They are people just like you and me, and can still find and enjoy the beauty in life.
There's a lot of debate among the special needs community regarding how to phrase the condition with the person. For example, if you Google how to describe someone on the Autistic Spectrum, there are debates between whether "my child is autistic," or "my child has autism" is more appropriate. A great article by Alex, who is on the autism spectrum, states that it's all up to the person.
And it's true. No matter how you refer to someone in the special needs community, it's most important to respect how the individual wants to be referred as. I believe as long as you have a good intention, and are open to accepting and respecting how to talk about a person, then you're doing a great job. Even as Alex alludes, there will be people who are going to be offended no matter what you use. The most important thing to remember is to be respectful and follow the wishes of the person who you are describing.
As for Down Syndrome, I don't mind if you say "with Down Syndrome," or "who has Down Syndrome." As long as you're using positive connotation and dropping anything that's negatively associated, I'm content with whichever.
Lastly, Down Syndrome is not a spectrum. Each individual who has it is so different.
It's best to describe someone appropriately to their own needs. When I talk about my brother, I let people know that he's a little more on the severe side of Down Syndrome in terms of his communication, as he is nonverbal. I always follow up that even though he's nonverbal, he's excellent at expressing his love and kindness towards others.
As always, thanks for reading! Remember, please don't shy away from asking any questions-- always better to ask rather than sit in ignorance. And especially when it comes down to how to refer to someone.