Trading when the NYSE opens is definitely not the best time to start a trade. It's the most volatile and is prone to lots of false moves. One thing that sometimes works is if the 2nd to last hour is down, there's a good chance that the first half of the last hour will be down as well, and if both of these time periods are down there's a very good chance the second half of the last hour will be down as well.. and it works upward as well.. Of course it doesn't always work that way but we work in percentages of probability of success so finding a pattern like this is a good thing.
Also, sometimes Forex price action ignores equity market activity altogether..
but there is no "best time" to trade.. It would depend on your style and what you hope to accomplish. I start trades based on milestone events in the price action such as breakouts.. They can happen at any time.