Fact Check
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This claim is mostly untrue in the context of the systems on the ballot in BC’s referendum.

Summary:

1. Proportional representation does not lead to more voters choosing far-right parties.
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   Some research (Carter) shows voters may be more likely to vote for far-right parties with winner-take-all systems. Other research shows no effect of the electoral system.
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2. The claim that “extremist” politicians – or far-right parties – have “no chance” with winner-take-all systems like first-past-the-post is simply untrue.
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   The success of Donald Trump, Doug Ford and the Reform Party point to success the far-right can achieve with first-past-the-post, particularly when they can geographically concentrate their support.
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3. Small “far right” parties in Europe do win more seats on average with proportional systems than with “winner-take-all” systems.
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   Proportional representation makes it easier for voters for smaller parties in general to achieve representation, but how much easier depends on the design of the system.The systems on the ballot for BC are all “mixed” systems – which don’t tend to result in any more parties with seats than Canada currently has (see Fact Check on number of parties with seats and fringe parties).

   Research by Pippa Norris (cited by opponents of proportional representation) looking at 39 countries over 14 years shows that under majoritarian systems, with 8.6% of votes, far right parties won 3.5% of seats, or a ratio of 0.40. Under combined or ‘mixed’ systems, with 4.4% of the vote far right parties won 2.1% of the seats, or a ratio of .48 (slightly more seats with mixed PR).

Far-right Parties, Representation and Power in Electoral systems

Proportional representation makes it easier for voters for small parties to achieve representation – in proportion to their support. This naturally includes parties of different ideologies. When opponents of proportional representation speak of “extremists”, though, they usually mean the “far right”, not the “far left”.

Pippa Norris’s (2017) study of elections in 39 countries over 14 years found that far-right parties were the most successful at winning seats in highly proportional systems (such as List PR systems not on the ballot in BC) and least successful in winner-take-all systems – with mixed systems (such as those on the ballot in BC) just slightly better for electing far-right parties than winner-take-all systems.

However, she notes that Canada’s Reform Party was very successful because Canada’s geography meant with first-past-the-post they could concentrate all their support in one region.

Representation is not the same as influence or power.

Sometimes far-right parties are included in governing coalitions, and sometimes not, depending on their policies and how much support those enjoy with the other parties and the public.

Three examples:

In Austria, a far-right party with 26% of the vote (not a fringe party) is in a majority coalition government with a centre-right party (which got 31.5% of the vote). With this level of support for their shared policies, it is hard to argue that these parties would not also have done well in winner-take-all system.

In Sweden, the Netherlands, and Germany, the other parties in the legislature (making up over 80% of the vote) have chosen to cooperate to deliberately prevent their small far-right parties from being in government or having any power over policy.

In New Zealand, a “populist” party – New Zealand First – is in a stable coalition government with the Labour Party (with a supply confidence agreement with the Green Party – three parties representing a majority). New Zealand First was also in a coalition government with the Labour Party in 2005.

New Zealand First leader Winston Peters unquestionably has a strong anti-immigration position. However, his other policies are more difficult to fit into a “right wing” box, including centrist and left wing economic policies such as opposing privatization of state services, enhanced benefits for senior citizens, nationalizing banks, writing off student loans, and increasing minimum wage to $20 an hour. Thus, characterizations of New Zealand First as a “far right” party are not correct. As the smaller party in the agreement, New Zealand First was not able to see many of its policies enacted.
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Voting Patterns in Proportional and Winner-Take-All Systems

More people do not vote for far-right parties with proportional systems. The factors that lead to people choosing to vote for far-right parties are complex. Here is what the research shows:

• A study looking at 33 right wing extremist parties over 23 years found: “While proportional electoral systems do undeniably make it easier for extremist parties to gain legislative representation, there is absolutely no evidence to suggest that they promote extremism. Instead, the share of the vote going to extremist parties appears unrelated to the type of electoral system employed.” (Carter, 2002)
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• A study looking at 13 anti-immigrant parties over 10 years found “the effect of proportional representation turns out to be not significant” (Van Der Brugh, 2005)
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• A study looking at Austria, France, Belgium, Norway, Germany, Italy and Denmark found: “We can see that the coefficient for the disproportionality of the electoral system is in fact positive, rather than negative as was anticipated. That is, the odds of voting for the extreme right actually increase as the disproportionality of the electoral system increases.” (Carter, 2006)
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References

Arzheimer and Carter (2006). Political opportunity structures and right wing extremist party success. https://onlinelibrary.wiley.com/doi/full/10.1111/j.1475-6765.2006.00304.x

Carter, E. (2002) Proportional Representation and the Fortunes of Right-Wing Extremist Parties. https://www.tandfonline.com/doi/abs/10.1080/713601617

Norris, Pippa. Does PR promote extremism redux? https://www.academia.edu/2748982/Does_PR_promote_political_extremism_redux

Van Der Brugh (2005). Why some anti-immigration parties fail and others succeed. http://journals.sagepub.com/doi/10.1177/0010414004273928