Only you can answer your own question because everyone has a different comfort level. I was in Istanbul in September (3rd trip traveling alone) and can tell you it is a huge, spread out city (albeit the "key" tourist sites are concentrated in Sultanahmet, the "Old City"). Another area where there are large crowds because it's a transport transfer point is "Taksim Square". The one safety precaution I would follow is stay away from any demonstrations or huge gatherings. If you want to travel low-key, you can avoid tourist areas altogether by booking a hotel on the Asia side which is mostly local (Kadikoy has nice hotels, for example)...there are also plenty of local neighborhoods right by the airport (although there is no evidence that tourists areas are targeted, and you will see a lot of tourist areas patrolled by cops with guns). Frankly, you have more risk in a car just driving to the airport than any terrorist attack, so keep that in mind.
No one has claimed responsibility so I would not assume one way or another who is responsible or what their motivations are. It's clear that the government is in limbo right now because Erdogan cannot put together a governing party after losing the majority in Parliament. That leaves an opening for upheaval and some group took advantage of that.
Keep following the news because you'll get a sense of what the government's reaction will be in the coming days/weeks. I would guess that areas will be more heavily patrolled.
Also, it seems that close to 100 people were killed (not 50). I am not sure the confirmed the final number yet, it may be higher.