I decline to comment on the "health" part of your question due to the extremely controversial and often unscientific public debate about the human health impact of non-ionizing electromagnetic radiation. However, your question does pose several good technical questions that are unrelated to health, so I will attempt to answer those without reference to biology/physiology.
Signal strength, as a concept, is intuitively defined as the amount of electromagnetic energy which was originally transmitted by the base station that is received by the receiver, compared to the amount of background "noise" (radio spectrum transmissions picked up by the receiver which do not originate from the transmitter).
So if you have a strong transmitter pushing out a lot of signal, but there's a lot of noise in between the transmitter and receiver, you get bad "signal strength", because it's hard for the receiver to clearly pick up the transmitter's signal over all the noise. Think of this like trying to verbally communicate with somebody when you are talking a few feet away from a running jet engine.
On the other hand, if you have a weak transmitter and very little background noise (i.e. the external environment is fairly "clean" of unwanted transmissions), it might be very easy to communicate even with low transmission energy. This would be like having a whispering conversation with somebody in an empty library with the air conditioner off; you both can understand each other perfectly despite speaking very softly.
The different 802.11 protocols don't particularly say "you can transmit at higher transmission energies" (which seems to be your question), but the newer protocols do devise smarter ways to use more of the electromagnetic spectrum to transmit more data, regardless of what the transmission energy is.
My advice to you would be to not worry about the "B/G/N" dichotomy if you are worried about the amount of transmission power. Instead, your goal should be to reduce the outside noise entering your wireless environment, and to make the base station (access point) as close as possible to your wireless client devices. These steps will make the devices themselves automatically and voluntarily reduce their transmit power, because, like two people talking in a quiet library, they'll be able to hear each other perfectly well without a lot of background noise.
To do that, you will want to make sure that there is a clear line of sight between your transmitter and receiver, and that you live in a place where there isn't a lot of external "chatter" on the wireless 2.4 and 5 GHz bands, such as the EM emitted by microwave ovens, cordless phones, and other wireless networks not controlled by you.
Alternatively, you can set up physical barriers that reduce or eliminate external radio frequency transmissions from entering your premises. I will not speak about the health impact of doing so, but I can factually state that setting up such a device (such as a Faraday cage) can reduce or eliminate other WiFi/bluetooth/microwave/cordless phone chatter in your environment, and this may make your WiFi more reliable and faster, while simultaneously reducing transmission strength.