Currently, there is no train and air bwteen Beihai and Liuzhou. On the other hand, the buses are also limited everyday. So, you may consider at Nanning.
Starting from Beihai early in the morning, you could catch a bus from Beihai Bus Terminal to Nanning in 3 hours. The buses are available from 06:00 to 21:00 with an interval of 20 minutes.

From Nanning to Liuzhou, there are frequent buses and trains for your choice. Both of them will take you about 3 hours on the way.

I am sure there is a bus directly from Beihai to Liuzhou. It is a 4-hour way.
There used to have a direct train from Beihai to Liuzhou years ago, but now, I am not sure. It seems only one train to Nanning, and you can take another train to Liuzhou.
Beihai to Nanning 3-4hrs 50-60RMB by train
2-3hrs 50RMB by bus