How does the speaker's husband feel about her writing?

It's clear that the speaker's husband, John, thinks her writing is silly, unhelpful, and worthless, even that it's a weakness.

Although we know that the speaker's writing is something she enjoys, something that gives meaning to her life and a purpose to her days, the husband is dead-set against it. He's a physician, and he claims that she needs to rest in order to get better and get over her anxiety and depression.

Of course, that's terrible advice. Writing could make the narrator tired, of course, but it could also give her a welcome distraction and a way to channel her feelings, which would help her get better.

But regardless of how ill-conceived and unhelpful John's views are, let's find some places in the text that reveal them.

First, the narrator confides this:

"So I take phosphates or phosphites—whichever it is, and tonics, and journeys, and air, and exercise, and am absolutely forbidden to 'work' until I am well again.

Personally, I disagree with their ideas.

Personally, I believe that congenial work, with excitement and change, would do me good.

But what is one to do?

I did write for a while in spite of them; but it DOES exhaust me a good deal—having to be so sly about it, or else meet with heavy opposition."

The speaker means that it's her husband who has forbidden her from working, and writing is the work she longs to do.

Let's see some more evidence:

"He says that with my imaginative power and habit of story-making, a nervous weakness like mine is sure to lead to all manner of excited fancies, and that I ought to use my will and good sense to check the tendency. So I try."

Here, the speaker tells us that John believes that her ability to tell stories and be imaginative will actually make her anxiety worse. 

Finally:

"I am sitting by the window now, up in this atrocious nursery, and there is nothing to hinder my writing as much as I please, save lack of strength.

John is away all day, and even some nights when his cases are serious."

She means that there's nothing getting in the way of her writing while John is out. The implication is that John will force her to quit writing if he catches her doing it.

So, although it's sad and unhelpful, and though it's definitely a major factor in the narrator's eventual mental breakdown, John's opinion of Jane's writing is that it's a hindrance and a silly, unnecessary thing.