I know prednisone lowers your immune system, so can I take Benadryl (diphenhydramine) instead?

Yes, in occasions. You've got an excellent point. For example, in treating sinusitis caused by allergic nasal congestion, Benadryl (diphenhydramine) treatment has much less side effects than prednesone.
Sure. Prednisone is generally reserved for severe allergy symptoms or asthma attacks. For nasal allergies, instead of Benadryl (diphenhydramine) i'd recommend a longer acting, less sedating antihistamine like Cetirizine or fexofenadine. For asthma, antihistamines won't help.
Short course OK. Your body makes a stress hormone very similar to Prednisone every day. A short course (5-7days) is often used to assist your own stress hormone response to get things back in order quickly & avoid prolonged symptoms. Prolonged use can suppress the immune system & should be avoided. Antihistamines can be beneficial if the issue is histamine based but will replicate all the benefits of the steroid.
Wrong question. Why taking prednisone? Dose and schedule? Duration? Benadryl (diphenhydramine) helps no skin conditions except histamine reactions, mostly hives. And for that there are much better antihistamines. So they are used in different circumstances. Benadryl (diphenhydramine) is a sleeping pill. Take only at bedtime. You need to indicate the reason for prednisone before any further discussion.