"Common sense" is sound/reasonable judgement. ("Should I wear my bikini in the snow?" -- "Of course not! Use your common sense!") "Step by step" is in a logical sequence or progression, possibly with the idea of gradual/small increments forwards. Both "common sense" and "step by step" are being used to modify the noun "basis". "On this basis" = "in this way" ---> in a reasonable and steady manner

It might be easier to understand this clause by parsing it as follows: "to try to figure out" | "how we best move" | "on a common sense, step by step basis" | "to address this very important issue." In other words, the sentence means "try to find the best way to address this important issue by approaching it in a step-by-step and common-sense manner." "On a common sense, step by step basis" is an adverbial phrase that modifies the verb "move."

I do not understand the phrase "move on common sense", but common sense means basic smarts or intelligence. Like for example, locking your front door when you leave home so as to avoid getting robbed is an example of common sense. To do something on a "Step by step basis" means that you do it gradually. Not all at once, not quickly.

move on a common sense basis and a step by step basis. Common sense means doing what most people think is the right thing to do. If it is cold outside it is common sense to wear a coat. If your car lights are broken it is common sense to repair them. Sometimes people say things are common sense when it is just their opinion. If a child is naughty it is common sense to hit them. Maybe some people think this is true, maybe others don't. A step by step basis is to do things in order. Do one thing before you start another. Don't try to do too much too quickly.