It's really a matter of parenting scientists, but this will appear on Father's Day. That is, suppose you're a parent and would like your child (children) to appreciate science. Not that they should become scientists -- I don't believe that this is the only career to pursue even though I did do so. (I also had my time pursuing engineering and computer science. I ended up going with the science, but it was a real question up through completing my undergraduate degree and receiving an engineering job offer.)
In this, I'll largely be drawing from examples my mother set. A good parent learns from all good parents, not just those of their own gender.
As a parent, perhaps the greatest thing you can support in your kids is the idea "The universe is a very interesting place." You don't need to know all the science yourself. It might even be a benefit if you don't. In any case, it's a family article of faith that children are born scientists -- investigators of the universe around them. When they start asking about why the sky is blue, grass green, ants are going that way, clouds are puffy rather than stringy ... cheer. You don't need to know the answers yourself. It doesn't take many years before your kids are asking questions that nobody knows the answer to. So, maybe answer where you know the answer. Where you don't, one or more of: "That's interesting. I don't know the answer. What do you think it is? How could we figure it out? Where would we search on the net?" is wonderful. The universe is incredibly interesting. If you're no great fan of insects, and find yourself parent (or, in my case, uncle) to a child who thinks they're incredibly interesting, cheer their interest. Insects are very interesting, at least to folks who study them, or maybe found fireflies pretty, and so on. In any case, I learned from my mother that you don't need to share the exact interest, and certainly don't need to know a lot about that particular thing*. The universe is interesting. That includes the parts you're not enthusiastic about yourself.
Second major thing is -- not all ideas will work out, and that's ok. "You'll think of more ideas later, and they'll be even better." Abandoning ideas that don't work out, I mentioned earlier, is one of the central skills of a scientist. If the idea doesn't work, don't spend your time 'reassuring' your child that it's 'ok that you failed'. They didn't fail, period. They successfully carried out one of the most important things a scientist can do -- test an idea and realize it doesn't work. Another better response is "Ok, since this doesn't quite work, what do you think might work instead?" Almost every idea that a scientist has, doesn't work. I doubt this is unique to science, so whether your child goes anywhere near science it's a good skill to have -- to have your idea not work, and to take it in stride and move on to the next idea.
I didn't even remember the example myself, but in my 20s or so, mom mentioned my 'blood theory of life'. I was 4 at the time, she said. The theory was, "We all have a finite amount of blood. When it runs out, say because of a trauma like a car accident or a blood disease like leukemia destroying healthy blood cells, you die." (I did know about leukemia at that age.) I don't exactly remember the responses (hey, I was 4, and 4 was ... several ... years ago), but I'm positive it wasn't "my what a wonderful idea". It was, I'm sure, in the vein of "That doesn't work because bone marrow makes more blood cells, so the supply isn't limited." Far better answer. Told me the idea didn't work, and told me something new that I could then think about more. Wow. There's even more to learn about the universe! And, it is possible for me to learn it!
This is another major skill to encourage in your children. "You may not know or understand something now, but you can learn it." Maybe they have to devote some time or effort, but it can be done. Things are possible, and regardless of where you are today, in time you can get where you'd like. Again, nothing unique to science about this. And, for that reason, a good reason to encourage it with respect to science. The skill, learning more stuff, is needed for almost everything. In science, finding out more about the universe -- that your child finds interesting -- the rewards follow more immediately from the effort than might be in other areas. In sports, if your child loves any healthy sport, support that. Skills can transfer, most especially the skill of training to do their sport well. Later, when some other sport becomes the greater interest, this overarching skill of training to do better will transfer. Same for study, practice, learn in science transferring to any other academic area. And, though I haven't seen it as often, between the sports and academics.
*I'll embarrass mom a little. She knew, and knows, far more than she has ever laid claim to. But, the key is regardless of how much she knew, she could never have hoped to answer all questions I asked. Same way that regardless of how much I know, I could never hope to answer all the questions my daughter has asked. As a parent, for us both, the key was, help your child learn more. Encourage that desire to learn more. Support the skills to learn more.
Additional support doesn't need to be expensive. Go for walks, look at the sky (works both day and night), visit the library, tour the internet with your child (perhaps at the library). An inexpensive telescope or microscope, or rock kit, or bug collecting kit, and so on can all be nice additions. But they're not necessary. If you do get things like this, it's a good idea to match them up to your child. A child interested in the stars will be thrilled with a reasonable telescope, while a child interested in insects ... not so much. Go with the flow here, a matter of good parenting anyhow.
Chances are that your child won't become a scientist. That's fine. What they're very likely to do, however, is become interested, informed people. And that's not only fine, but vital. And you'll get to spend time with them, and hear them being enthusiastic about their latest discovery for years while they grow up.