There is no particular ratio at which the engine becomes a 'high' bypass, though the it is generally taken to be around ~5:1. The reasons are more historic than technical.
The first large scale development of the high-bypass turbofans was spurred by the USAF's CX-HLS competition, which lead to the C-5 Galaxy. The requirements of the competition (range and fuel efficiency) lead to the development of a new generation of engine by GE and PW, which developed the TF39 and JTF14E respectively. GE won the contract with the TF39 engine, which had a previously unheard of 8:1 bypass ratio.
PW figured out that it better develop a high bypass engine itself or lose the market and developed the JT9D, which had a 5:1 bypass ratio. This engine was the first widely used engine with high bypass ratio (the previous engine had bypass ratios in the range of ~1 or so) and necessitated a special mention, which came to be called high-bypass engines, once it became widely used, especially in the Boeing 747 (Ironically, it was the two losers of the CX-HLS competition PW and Boeing, which won the immediate race in civil aircraft).
Joe Sutter acknowledged the effect of the CX competition in the development of 747:
I should add that fostering large high-bypass engines was all that the USAF C-5 competition contributed to the Boeing 747, as my new airplane would be called.
Not to be left behind, GE developed the CF6 series with bypass ratios of 6:1 and RR developed the RB211-22 which had a bypass ratio of ~5:1. Due to these, the ratio of around 5:1 (or >4:1) is usually taken as the dividing line between the 'high' and 'low' bypass ratio turbofans.