The crossbench peer said she had been accused of being “responsible for people dying in pain”.
She is among those expected to speak in a historic debate on Friday as the Terminally Ill Adults (End of Life) Bill returns to parliament.
Baroness Grey-Thompson said she is not opposed to the principle of assisted dying but is against the legislation in its current form.
The peer said the emails have been tough to read, but she accepts it’s “part of the job”.
She said: “I’ve had emails, people saying, ‘thank you so much for protecting us’. And I’ve had some incredibly abusive ones as well, saying that I’m responsible for people dying in pain, and it will be on me.”
Baroness Grey-Thompson said the experience of losing her own mother to cancer means she understands people’s strong feelings, but insisted there should be more focus on improving end-of-life care.
“It’s absolutely appalling that people are scared about how they’re going to die. But we should be having proper specialist palliative care,” she said.
On the abusive emails, she added: “If you accept the role in the House of Lords, you have to accept there are a lot of people who don’t like what you do.”
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Around 200 peers have put their names down to speak on the bill when it comes to the Lords on Friday.
It is thought this second reading stage could go into a second day to allow everyone to have their say on the proposed legislation.
The Commons has voted twice on the overall bill, having approved the principle of assisted dying. Peers in the Lords will now be given the chance to put forward changes.
The legislation, which applies only to England and Wales, would face further scrutiny at committee and report stages. Those in favour are hopeful it could pass into law by the end of this parliamentary session in spring.
But there would be a four-year implementation period before an assisted dying service would be up and running.
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Baroness Grey-Thompson said: “The bill could still fall. It could run out of (parliamentary) time.”
Former lord chancellor and justice secretary Lord Charlie Falconer, the sponsor of the bill in the Lords, said he was “very positive that the bill will pass”
Assisted dying campaigner Dame Esther Rantzen has previously urged members of the House of Lords not to block landmark legislation on the issue.
Labour MP Kim Leadbeater, who introduced the bill to the Commons last year, appealed for “no funny games” in terms of procedure in the Lords.
She said she “would be upset to think that anybody was playing games with such an important and such an emotional issue”.
Baroness Grey-Thompson added: “It’s my intention to bring a number of amendments to make it as safe as it can possibly be. But we do need to have a really detailed discussion.”
