I just rented a home in that area that I have managed for close to two years and I have never had an issue. The neighbors are friendly and (in our case) despite being relatively close to the old water and sewer offices and elementary school the traffic is quite minimal.
Hope that was helpful. You may want to check with the City of Miami's website for crime statistics from the Police dept. I believe you can get this online.
Some areas of Coconut Grove have private policy or security services like the Bonita Park area where roving guards police the area 24 hours a day. I've been renting a summer home in the area for the past 4 years and never had an issue with crime during that time. The rest of the year the property is also occupied by its owners and they have not had issues either since they bought the property. They are very happy with the security of the neighborhood. Checking with the local police department should be your next step if you wish to inquire about crime rates in the area.
Southwest Grove is a beautiful place to live. We just sold a house there and met many of the neighbors. They are very friendly and community oriented. Let us know if you need help finding your dream home.