The national Constitution Party website has reposted an excerpt and link from the Montgomery Advertiser’s editorial against Alabama’s restrictive ballot access laws:
Alabama’s Ballot Access Laws Too Restrictive
Voters often complain about what they see as a lack of meaningful choice on the ballot, about rarely having the chance to vote for someone other than a Republican or a Democrat. Because Alabama has some of the most stringent ballot access laws in the country, that’s not likely to change without an overdue revision of the rules.
The same Montgomery Advertiser article, along with a similar one from Oklahoma, has found its way to the national Libertarian Party facebook page and twitter, both of which I contribute to, via a prior IPR story. Excerpt from the prior IPR posting:

In IPR comments, Ballot Access News editor Richard Winger wrote:
Alabama, in last month’s election, had a U.S. Senate candidate with only one candidate on the ballot. The Democrats didn’t run anyone for U.S. Senate, and no minor party or independent candidate could get on the ballot. Even for Alabama, the specter of a one-candidate election for such an important office looks pretty bad. Also in 2014 Alabama was one of only five states with a Dem-Rep ballot monopoly for all the statewide offices. These are facts that weren’t available in previous sessions. Also it is worth noting that Alabama voter turnout in 2014 was shockingly low.
http://www.electproject.org/2014g is the link for 2014 turnout data. Alabama’s turnout was 32.6% of the eligible electorate.
….
The Missouri legislature in 1989 was incredibly hostile to ballot access reform. Our bill was presented in committee in the Senate. After the witnesses in favor spoke, Senator Webster said, “Only fruits and nuts are interested in this.” And no one else said one word. No member of the committee moved the bill. But, we kept re-introducing it, and finally in 1993 it was signed into law. The LP hasn’t had to petition in Missouri ever since. The Constitution Party is also safely on the ballot there.
And:
26 states have made it easier for a party to remain ballot-qualified in the last 30 years: Alaska, Arizona, Arkansas, California, Colorado, Florida, Georgia, Hawaii, Louisiana, Maine, Maryland, Massachusetts, Michigan, Minnesota, Montana, Nebraska, Nevada, North Carolina, North Dakota, Ohio, Oregon, South Dakota, Texas, Utah, Virginia, and Wyoming. We made all this progress not by lawsuits, but by asking state legislators to do this. We are insane, crazy, nuts, if we don’t continue doing this. And now is the time when legislators are deciding which bills to introduce. Many states have strict deadlines for introducing bills.

Disclaimer: I (kind of) live in Alabama, and have been on the state exec comm of the LP here on and off since 1998. In 2012-2014 I was LNC alternate for a region that included both Alabama and Oklahoma. As a petitioner, I have worked on ballot access in Alabama many times, and in Oklahoma in 2000. I’m currently state director of Alabama for Our America Initiative and co-deputy director of the OAI Southern Region, which includes both Alabama and Oklahoma. One of the projects Alabama for OAI is working on is easing ballot access reform:
Our America Initiative supports ABLE (Alabama for Ballot Access Law Equality), a coalition of groups and individuals seeking an end to unfair signature requirements for candidates running outside the establishment parties. This year Alabama was one of only four states in which voters could choose only between Democrats and Republicans for statewide office. Third party presidential candidates could appear on the ballot only as “independents”, forbidden from choosing their preferred party label such as Green or Libertarian. Failure to earn 20% of the vote requires the gathering for the next election of signatures equal to 3% of the vote for Governor – that’s around 40,000 valid signatures statewide, around 6,000 valid signatures for a Congressional race, and around 400 valid signatures for a State House district. This signature threshold is the second highest in the country for statewide office. The good news is over half the states have eased ballot access over the past few decades. Nearby states including Mississippi, Louisiana and Florida have some of the easiest ballot access laws in the country. With 60% of voters wanting a third party, it’s clear why Republicans and Democrats are facing opposition to maintaining their barriers against election competition. Unfortunately, with a lawsuit recently dismissed alleging how burdensome Alabama’s signature requirements are, we urge Alabamans to email their senator and representative to pass Senator Ward’s pending bill to halve the signature requirements to 1.5% of the gubernatorial vote. And we hope everyone donates to Our America Initiative’s national lawsuit to open the debates to all viable candidates.
See also:
Alabama Ballot Access Bill Will Be Introduced Again
The Constitution Party of Alabama has been heavily involved in past efforts to reform Alabama’s ballot access laws and its chair, Joshua Cassity, started ABLE (Alabama for Ballot Access Law Equality) with help from Libertarians and others.

In Alabama I would say restrictive ballot access laws are definitely a part of the problem for sure.
I think the CP’s problem isn’t so much restrictive ballot access laws as it is their very narrow appeal and not-so-effective organizational structure & leadership.