An antenna does not directly influence power consumption. It is simply a passive resonating piece of wire that is "open-ended", hence current cannot flow through it. The power used for transmitting over an antenna therefor depends solely on the circuit trying to send, e.g. the transmit power.
In practice though, wireless circuits are trying to be smart, and they assume that if they sense a weak signal, it is weak not because the antenna is wrong or detuned, but because the wireless peer is far away or blocked. Thus senders often increase their transmit power if the signal is weak in the hope that this way they'll be able to bridge the distance (or get better throughput). So it can be said that a good antenna can reduce power consumption in comparison with a bad one, but this depends entirely on the circuit behind it.
Antenna design is complex, and in general nobody will be able to give a formula to describe signal-to-noise ratio or frequency based on antenna parameters. This is only possible in the simplest cases, and in all others, experts use simulation software to just estimate expected antenna performance. There are some very generic constraints (rule-of-thumbs) though, for example the lower the frequency gets, the larger the antenna you'll need, and other minimum lengths and gaps to be observed (for example those relating to the lambda/4 rule). But the performance of the antenna depends on many other things than just its own geometry, like for example distances to other objects in the vicinity, the materials of those, the geometric form, and even how you get the signal to the antenna (a wrongly matched impedance will loose most of your signal before it even gets to the antenna). Even how the same antenna is oriented in space makes a difference (polarization).