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Even there was some hurdles, i feel relatively safe in Beirut. Sure it is all depends, especially by region, but touristic places are usually very safe. Because of civil war in Syria, and enormous amount of refugees (who are often suffering from lack of money) it is better to take precautions.
At day it is completely safe, at night it is better to not walk on places where is no other people present.
As general in middle east countries, which is not related to me, but for woman better to travel with companion. Even in touristic places they are absolutely safe, if she is with companion, strangers wont leave some unpleasant comments about lonely lady.
Also... sometimes don't listen to locals who are scaring visitors, e.g. some people from jounieh telling scary stories about Nabatieh, just because they never went there (christian vs muslim area), but i did visited such areas and really enjoyed it, lebanese are friendly at south as well.
Only one thing - definitely avoid border areas, where is refugees and smugglers operating.
MAYBE (not sure) Tripoli and more north as well. Just because of Syria issue, they are not absolutely safe. However, maybe i am wrong...

That's a great question...do I feel safe in Lebanon? I've been here just under a month initially I was too overwhelmed by the majnoon driving & trash to think about safety! We started out in Tripoli and I went out a few times by myself during the day with very limited knowledge of the Arabic language. I felt completely safe. Night time is a different story especially during Ramadon. I didn't always feel safe with certain more aggressive refugees asking for money & following us. I never went out at night alone. Most of the people in Tripoli are lovely and very welcoming. I felt safer in Beirut no matter what the time, again I don't go out alone at night. I'm mostly with my husband and/or friends. I look forward to exploring even more of Lebanon!