There is a broken (open) part from an abestos pipe cover in my basement. It's near the staircase& a closed window. Should I have it removed or leave it?

Remove it. Any open sources of asbestos should be removed by people qualified to do so. That stuff tends to drift in the air as it falls off due to the process of deterioration when it is not covered. Don't try and remove it yourself and if it takes more than a very easy effort to temporarily cover it, just leave it alone and get someone that knows what they're doing to do the job for you and give you advice.