Short answer: You Can't
The Human Genome Project began "on a napkin" in 1984 and was declared complete in 2003. That's 19 years with access to overwhelming computational power. So, no, you can't manipulate DNA without computers.
Long answer: What's a computer?
Some years ago I coined a phrase, "technology dichotomy." The phrase describes the occasional author who wants, for example, time travel without first inventing the wheel. In your case it's the development of advanced bio-engineering without computers. The reality of technology is that you can't progress from one point to another without a vast pyramid of knowledge extending out below it.
DNA was discovered using x-rays. Radioactivity (from whence x-rays come) comes from the study of material physics. As does, among many other things, electronics. To justify no electricity-driven computers on your world you would basically need to justify no electricity ... except that electricity is a fundamental component of life and there's always some kite-and-key-weilding goombah trying to figure out how something works for no better reason than it's there. But that ignores the idea of "computers" generally.
Let's look at reality: as you understand more things, the foundation math gets more complex with it. As the complexity of math increases, the need to crunch it faster than a pencil and paper will allow increases. End result: a computer. But wait....
Most people don't realize the etymology of the word "computer" comes from the 1600s (yup, the sixteen-hundreds), meaning "one [a person] who calculates." In the beginning all computers were people. Then came mechanical computers like the Babbage Engine, then came electrically-driven mechanical computers, then (FINALLY!) came electronics.
It's impossible for an intelligent species to develop accounting without the need to expand to something faster than pencil and paper. Why do I say accounting? I can't prove it, but I'll eat my hat if the reason for the first computer (person) wasn't money. Some greasy politician wanted to total up his
bribes, I mean take, I mean benefits. And computers were born. (Seriously, it was undoubtadly the Rothschildes... you know it was the Rothschilds.... If you think someone's out to get you, you're probably right....)
Therefore, the answer is "no, you can't do it without a computer, but what's stopping your species from developing computers?"
Bear in mind that it's unlikely that a society that is biologically based wouldn't understand electron transfer and solid-state physics. They may find a way to do it without silicon, but they'd still have computers.