Babies who are under a year old should not sleep with any pillows, blankets or other soft objects in the crib or bassinet with them as the pillows, blankets and any other soft objects pose a suffocation hazard to the babies under 1 year of age.
Babies die of SIDS in cribs and bassinets when they sleep with pillows, blankets and other soft objects when they're under a year old.
Once the baby gets to 1 year old or older they can start sleeping with blankets and pillows and soft objects and should be just fine.
I would personally wait until the baby is at least 13 months old before letting them sleep with blankets and pillows or other soft objects just to be safe.
You can keep your baby warm with footed pajamas during the first year of life and then after that they can sleep with blankets and pillows.