Can I donate blood, plasma or bone marrow with malaria history?

How long ago? This depends on how long ago you had malaria and whether or not you were adequately treated. If you have traveled to a region known to harbor malaria, you are deferred for 1 year. If you lived there or have been diagnosed with malaria, you are deferred for 3 years adn must remain symptom free.
American Red Cross. recommends deferring blood donations for 3 years if you have had Malaria.
If cured, yes. If it' s a history of malaria and one 1. Has been treated for the appropriate malaria type for the recommended period, 2. Risk of exposure after treatment is low, and 3. There are no signs or symptoms of malaria, then it's ok- however, institutional policies vary so check with the specialist appropriate for the tissue that is being taken.