Well as with any deck, or CCG you have to be flexible, both in playing style and in how you think.
I do not actually have a gowen deck, so I cannot actually comment on how to play one from that side, nor do I know what cards you get in the starter deck. In all actuallity I do not really want to give you advice since I probably hate a good gowen deck the most. There rank up stuff and low level creatures drive me insane, however I have played against gowen a few times and what has been suggested so far are the typical openings for gowen.
If you do the 1 monk, 2 karina, and 3 archer, then while it is true that karina cannot attack, if as you say someone plays a row kill and kills your monk then she can for that turn. However her most important function is the buffing. Other strategies I have seen, would be to play a brave soldier, and then use panther soul on him the second turn. A fifty life fifty damage guy on the second turn is damn annoying to deal with. Of course you would need to have those cards in your starter deck for that to work. Lots of people also go with salmanders for the second or third turn as has been said, they do direct damage so can kill weak ranged units or damaged units and they have a strong attack. You also need to consider knowing when to let a unit die so that you can trigger a soul skill, and in that sense you need to find the soul skill setup that works best for your play style.
Try messing with your deck if you have to, and testing it out in the training area, or similarly test out the strategies suggested there so you do not have to worry about your record if that is important to you.
As I said before you need the courage and flexability to try out new and interesting things.
By the way, can someone explain how the monks active skill works? I have seen that guy do anywhere from thirty to sixty damage with it.