The meaning of the word optimism (according to Dictionary.com) means:
a disposition or tendency to look on the more favorable side of events or conditions and to expect the most favorable outcome.
Therefore, to write poetry with an optimistic perspective, the poetry must prove that good always overcomes evil. In regards to determining this, a reader will offer a completely subjective interpretation of the poem. This means that not all readers come to understand a poem in the same way- one may read the poem in a completely different way than another.
This being said, one would simply need to see if they agree that the poetry of Robert Browning speaks to them in an optimistic fashion. If one does, then they would see Browning as an optimistic poet. If they do not, they would not regard Browning as an optimistic poet.
For example, in a critical reading of "A Woman's Last Word
", one could justify the poem as optimistic or not optimistic. In an optimistic reading, one would justify the poem as such by stating that the speaker of the poem does not want to fight with her lover. Instead, she wants him to go to sleep so as to find peace in sleep. If one justifies the speaker as wanting peace for the lover, through sleep, then they could consider the poem as optimistic (good (sleep) overcoming bad (no sleep).
Another way to look at the poem would be that the speaker does not want to deal with the problems that night. Instead, she would rather put them off for another day. If a reader does not agree with this action, they would not consider that the outcome of the sleep would result in good. Therefore, the reader could justify the poem as not being optimistic. Basically, the speaker wishes to ignore the problems and simply bury them.