"The process we live today is very serious."
Translation:O processo que vivemos hoje é muito grave.
I agree - I think this was supposed to be "the system in which we live today is grave"
Clearly the given English translation is nonsense, but "grave" would not be used with system. It is very commonly paired with "situation," actually a journalistic cliche. But processo is not static but something going on. Vivemos should be translated "we are experiencing" I think. I guess the meaning is something like "the course our lives are taking today is gravely disturbing."
I think it depends on the intent of the author. I think the intent was to describe our human condition in a negative light, in which case "grave" would be the better translation for "serious". If the situation were not negative, e.g."He is a serious student." or "Was she being serious or was she joking?" , then "sério(a)" would be appropriate.