Is it possible to complete Demon's Souls 100% if I play offline?

Atlus will be taking down down the Demon's Souls servers (North American ones, at least) on May 31st.

I haven't got around to finishing Demon's Souls yet but I want to achieve a Platinum trophy in the game. Will it be possible/feasible once the servers are offline? How will that affect me?

Yes, it is certainly possible.

None of the Trophies for Demon's Souls are based around online features of the game.

PS3Trophies.Org has a list of the trophies and information on how to achieve them which corroborates this.

It is actually easier to get all achievements when playing offline the whole time because online play tends to screw with your world tendency.